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SYNOPSIS
OF
LINEAR ASSOCIATIVE ALGEBRA
A RHPORT ON ITS NATURAL DEVELOPMENT AND RESULTS REACHED
UP TO THE PRESENT TIME
BY
JAMES BYRNIE SHAW
Professor of Mathematics in the James MiUikin University
A
WASHINGTON, D. C. :
Published by the Carnegie Institution of Washington
1907
CARNEGIE INSTITUTION OF WASHINGTON
Publication No. 78
Z^t Boti (gattimcxe {prtee
BALTIMORE, MD. , U. B. A.
CONTENTS.
iKTIiODUCrilON.
PAOB
5
Part I. General Theory.
I. Deliiiitioiis. ......
1 1. The cliaracteristic equations of a number.
I II. 'I'lic characteristic equations of the algebra.
Associative units. . . . . .
Sub-algebras, Redncibiiity, Deletion.
Dedekind and Fkobenius algebras.
IV.
\^
VI.
VII.
VIII.
IX.
X.
xr.
XII.
Scheffers and Peiuce algebras.
Kkoneckeu and WEiEKf5TRASS algebras.
Algebras with coefficients in arbitrary fields.
Dickson algebras.
Number theory of algebras.
Function theory of algebras.
Cirou]) theory of algebras.
Kea
algebri
IS.
XIII. General theory of algebras.
Part II. Particular Algebras.
XIV^. Complex numbers
Quaternions. .....
Alternate algebras. ....
Biquaternions or octonions.
Triquaternions and (^uadriquaternions.
Sylvester algebras.
Peirce algebras. ....
ScHEFFERS algebras.
Caht.\n algebras. ....
XV.
XVI.
XVII.
XVIII.
XIX.
XX.
XXI.
XXII.
Part III. Applications.
XXIII. Geometrical applications.
XXIV. Physico-mechanical applications
XXV. Transformation groups.
XXVI. Abstract groups.
XXVII. (Special classes of groups.
XXVIII. Differential equations.
XXIX. Modular systems.
XXX. Operators.
Bibliography
9
31
.3.5
40
44
48
52
56
58
60
68
72
79
80
83
87
91
93
101
107
HI
113
120
120
125
129
1.33
133
134
1.35
ERRATA.
Page.
11. Line 13, for \e,j\' read \(\j\\
15. In the foot-uotes change numbering as follows: for 1 read 2, for 2 read 3, for 3 read 4,
for 4 read 1.
26. Line 21, for A" read h^ .
33. Line 15, for AejC^ read ^e,f..
34. Line 6, for [?«, (,",) read [?«,'(;",).
49. Line 6, for ?h,"|i read ?w| + ,.
53, 54. In the table for r >6 in every instance change r—2 to r—3, and r— 3 to r — 4.
In case (27), hoioever, read e, = (311) — (12 r — 3).
57. Line 8, for t, read t^ .
59. Line 33, remove the period after A.
67. Line 12, insert a comma (,) after "integer".
68. Lines 9 and 10, cliange y to ■-•.
71. Line 17, in type III for e,, read e^.
72. Last line, for a q «~' read a q «"'.
73. Line 3 from bottom, for jk <■ read jk ' .
94. Line 7, for Srj' read SPj"'.
94. Last line, in the second column of the determinant and third line for S. /"' a j^ <p ~<p
read S .j'' nj'' <p .
100. Line 12, for (t> = <p read </> = 0-
106. Some of these cases are equivalent to others previously given.
107. Line 3 from bottom, /or e.^ = (221) read e^ = (211).
116. Line 25 should read p = '- ^r~^ — .
124. Note 3, add: of. Beez '-'.
128. Line 11, for i = I k^ read i = 1 ....//, .
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
INTRODUCTION.
This memoir is genetic in its intent, in that it aims to set forth the present
state of the mathematical discipline indicated by its title: not in a comparative
study of different known algebras, nor in the exhaustive study of any particular
algebra, but in tracing the general laws of the whole subject. Developments
of individual known algebras may be found in the original memoirs. A partial
bibliography of this entire field may be found in the Bibliography of the
Quaternion Society,^ which is fairly complete on the subject. Comparative
studies, more or less complete, may be found in Hankel's lectures,^ and in
Cayley's paper on Multiple Algebra.* These studies, as well as those men-
tioned below, are historical and critical, as well as comparative. The phyletic
development is given partially in Study's Encyklopildie^ article, his CJiicufjo
Congress''' paper, and in Cartan's Encyclopedic^ article. These papers furnish
numerous expositions of systems, and references to original sources. Further
historical references are also indicated below. **
In view of this careful work therefore, it does not seem desirable to review
the field again historically. There is a necessity, however, for a presentation
of the subject which sets forth the results already at hand, in a genetic order.
From such presentation may possibly come suggestions for the future.
Attention will be given to chronology, and it is hoped the references given
will indicate prioritv claims to a certain extent. These are not always easy
to settle, as they are sometimes buried in papers never widely circulated, nor
is it always possible to say whether a notion existed in a paper explicitly or
only implicitly, consequently this memoir does not presume to offer any authori-
tative statements as to priority.
The memoir is divided into three parts : General Theory, Particular Si/s-
temSy Applications. Under the General Theory is given the development of the
subject from fundamental principles, no use being made of other mathematical
disciplines, such as bilinear forms, matrices, continuous groups, and the like.
'Presented, in a slightly diflerent form, as an abstract of this paper, to the Congress of Arts and
Sciences at the Universal Exposition, St. Louis, Sept. 33, 1904.
'Bibliography of Quaternions and allied systems of mathematics, Alexander Macfarlane, 1904, Dublin.
'Hankbl 1. References to the bibliography at the end of the memoir are given by author and
number of paper.
*Caylet9. 'StudiS. 6Stcdt7. 'CabtasS.
•Beman 3, GiBBS 2, R. Graves 1, Haqen 1, Macfablane 4.
6 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
We find the first such general treatment in Hamilton's theory^ of sets. The
first extensive attempt at development of algebras in this way was made by
Benjamin Pkirce^. His memoir was really epoch-making. It lias been critic-
ally examined by Hawkes^, who has nndertaken to extend Peirce's method,
showing its full power*. The next treatment of a similar character was by
Cartan', who used the characteristic equation to develop several theorems of
much generality. In this development appear the scmi-simjjle, or Dedekind,
and the jJseiwZo-nu?, or nilpotent, sub-algebras. The very important theorem
that the structure of every algebra may be represented by the use of double
units, the first factor being quadrate, the second non-quadrate, is the ultimate
proposition he reaches. The latest direct treatment is by Taber", who
reexamines the results of Peirce, establishing them fully (which Peirce had
not done in every case) and extending them to any domain for the coordinates.
[His units however are linearly independent not only in the field of the
coordinates, but for anj' domain or field ]
Two lines of development of linear associative algebra have been followed
besides this direct line. The first is by use of the continuous group. It was
PoiNCARE^ who first announced this isomorphism. The method was followed
by ScHEFFERS*, who classified algebras as quaternionic and noii quaternionic.
In the latter class he found "regular" units which can be so arranged that
the product of any two is exj^ressible linearly in terms of those which
follow both. He worked out complete lists of all algebras to order five
inclusive. His successor was Molien®, who added the theorems that quater-
nionic algebras contain independent quadrates, and that quaternionic algebras
can be classified according to non-quaternionic types. He did not, however,
reach the duplex character of the units found by Cartan.
The other line of development is by using the matrix theory. C. S. Peirce^"
first noticed this isomorphism, although in embryo it appeared sooner. The
line was followed by Shaw " and Frobenius ^'\ The former shows that the
equation of an algebra determines its quadrate units, and certain of the direct
units; that the other units form a nilpotent system which with the quadrates
may be reduced to certain canonical forms. The algebra is thus made a sub-
algebra under the algebra of the associative units used in these canonical forms.
Frobenius proves that every algebra has a Dedekind sub-algebra, whose
equation contains all factors in the equation of the algebra. This is the semi-
simple algebra of Cartan. He also showed that the remaining units form a
nilpotent algebra whose units may be regularized.
It is interesting to note the substantial identity of these developments,
aside frojn the vehicle of expression. The results will be given in the order
of development of the paper with no regard to the method of derivation. The
references will cover the difi'erent proofs.
'Hamilton 1. 'B. Peirce 1, 3. 2IIawkes2. *IIa\vkes 1, 8, 4.
' CaKTAN 2. 'TaDER 4. ' POINCAHE 1. 'SCIIEFFEU3 1, 2, 8.
•Moi.iENl. '» C. S. Peirce 1, 4. " Shaw 4. "Frodbnios 14.
INTRODUCTION 7
The last cluipter of tlic general llioory gives a sketoli of the theory of
general algebra, placing linear associative algebra in its genetic relations to
general linear algebra. Sonic scant work has been done in this development,
particularly along the line of symbolic logic' On the philosophical side,
which this general treatment leads up to, there have always been two views
of complex algebra. The one regards a number in such an algebra as in
reality a duplex, triplex, or multiplex of arithmetical numbers or expressions.
Tiie so-called units become mere nmhrae serving to distinguish the dilTerent
coordinates. This seems to have been Cayley's^ view. It is in essence the
view of most writers on the subject. The other regards the number in a linear
algebra as a single entity, and multiplex only in that an equality between
two such numbers implies n equalities between certain coordinates or functions
of the numbers. This was Hamilton's'' view, and to a certain extent Gkass-
mann's.' The first view seeks to derive all properties from a multiplication
table. The second seeks to derive these properties from definitions applying
to all numbers of an algebra. The attempt to base all mathematics on arith-
metic leads to the first view. The attempt to base all mathematics on algebra,
or the theory of entities defined by relational identities, leads to the second
view. It would seem that the latter would be the more profitable from the
standpoint of utility. This has been the case notably in all developments
along this line, for example, quaternions and space-analysis in general.
Hamilton, and those who have caught his idea since, have endeavored to form
expressions for other algebras which will serve the purpose which the scalar,
vector, conjugate, etc., do in quaternions, in relieving the system of reference
to any unit-system. Such definition of algebra, or of an algebra, is a develop-
ment in terms of what may be called the fundamental invariant forms of the
algebra. The characteristic equation of the algebra and its derived equations
are of this character, since they are true for all numbers irrespective of the
units which define the algebra; or, in other words, these relations are identically
the same for all equivalent algebras. The present memoir undertakes to add to
the development of this view of the subject.
In conclusion it may be remarked that several theorems occur in the course
of the memoir which it is believed have never before been explicitly stated.
Where not perfectly obvious the proof is given. The proofs of the known
theorems are all indicated by the references given, the papers referred to con-
taining the proofs in question. No fuller treatment could properly be given
in a synopsis.
' C. S. PeIRCE I, 2, SCIIUOEDER 1, WHITEHEAD 1, RUSSEI.L 1, SHAW t.
« Caylet 1, 9. See also Gibus 1, 3, 3. 3 Hamilton 1, 3. ■'Grassmaks 1, 2.
PART I. GENERAL THEORY.
I. DEFINITIONS.
1. EARLY DEFINITIONS.'
1. Definitions. Let there be a set of r entities, e, . . . . e,, which will be
called qualitative units. These entities will serve to distinguish certain other
entities, called coorc^wjafes, from each other, the coordinates belonging to a given
range, or ensemble of elements; thus if a; is a coordinate, then Mj^j is dilTerent
from OiCj, if i ^J, and no process of combination belonging to the range of «(
can produce a^Cj from ajCj. Thus, the range may be the domain of scalars
(ordinary, real, and imaginary numbers), or it may be the range of integers, or
it may be any abstract field, or even any algebra. If it be the range of integers,
subject to addition, subtraction, multiplication, and partially to division,
then by no process of this kind or any combination of such can ajCj become
ttiCj. These qnalijled coordinates may be combined into expressions called
complex, or hypercomplex, or multiple numbers, thus
r
a ■=■ 'S. a^ gj
t = i
\i\ this number each «, is supposed to run through the entire range. The units
Cj, or le,, are said to define a region of order r.
2. Theorems : ^
(1) (r(-|-i)ei = aei + 6<?(, and conversely, if + is defined for the range.
(2) Ofij = e, = , if belongs to the range.
r
(3) 1 aiei = 0, implies 0^ = (i = 1 r)
1 = 1
(4) If 2 Oiff = S 6i<°i; then ai = 6,., t = 1 r, and conversely.
1=1 1=1
Theorems (3) and (4) might be omitted by changing the original definitions,
in which case relations might exist between the units. Thus, the units + 1
and — 1 are connected by the relation + 1 + ( — 1) = 0.
Algebras of this character have more units than dimensions.
3. Definitions. A combination of these multiple numbers called addition
is defined by the statement ,.
a + /3 = 2 (a, + h) e,
i = l
'HankelI, Whitehead 1. Almost every writer has given equivalent definitions. These were of
course more or less loosely stated.
' Whiteheab 1.
10 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
In quaternions and space-analysis the definition is derived from geometrical
considerations, and the definition used here is usually a theorem.^
4. Theorem. From the definition we have
a + (3 = i3 + a a + {(3 + y) = {a + i3) + y
when these equations hold for the range of coordinates. If subtraction is
defined for the range, it will also apply here.
5. Theorem. If m belongs to the range and if ma is defined for the range
(called multiplication of elements of the range) then we have
r
OT a = 2 (m a,) Cj
1=1
6. The units are called units"-, or Haupteinheiten'^ , and the region they
define is also called the ground^ or the hasis^ of the algebra. The units are
written also" (1,0,0, ), (0, 1, 0, ), (0,0, 1), the position of the 1
serving to designate them. The implication in this method of indicating them
is that they are simply ordinary units (numbers) in a system of «-tuple numbers,
the coordinates of each n-tuple number being independent variables. This view
may be called the arithmetic view as opposed to that which may be called the
vector view, and which looks upon the units as extraordinary entities, a terra
due to Cayley. There are two other views of the units, namely, the operator
view, and the algehraic view. The first considers any unit except ordinary
unity to be an operator, as (—1) or the quaternions i,j, h. The second con-
siders any unit to be a solution of a set of equations which it must satisfy and
as an extension of some range (or domain, pr field); or from a more abstract
point of view we consider the range to be reduced modulo certain expressions
containing the so-called units as arbitrary entities from the range. Thus, if
we treat algebraic expressions modulo i'^ + 1, we virtually introduce V — 1
into the range as an extension of it.'^
7. Definition. We may now build a calculus** based solely on addition of
numbers and combinations of the coordinates. This may be done as follows :
Let the symbol / have the meaning defined by the following equations : if
r r
a = 2 Oi ej ^ = 2 Xi €i
1=1 i=l
then
r
/ . a ^ = 2 UiXi
i = 1
It is assumed that the coordinates a, x, are capable of combining by an associ-
ative, commutative, distributive process which may be called multiplication,
so that UiXi, is in the coordinate range for every Oj and a-j, as well as XafCi.
"Hamilton 1, 2, Grasshann 1, 2, cf. Macpablane 1. 'Grassmann 1.
« Weiekstrass 2. «Taber1. SMoLiENl. «Dedekind 1, Bkki-ott 1.
iguAW 13. *See §21 for dillereuce between a calculus aud an algebra.
DICFIiNlTIONS
11
Evidently
1 . ri^ = x.
E=lcJ.e,^,
( 1
Also, if / ::}:./
/.f'.^'i=l 1.6(61=0
8. Theorem. We have
9. Definition. We suy thut a and ^ are orthogonal if 7 . a^ = 0. The units
g]. . . ., Cr therefore form an ortJto(jonal system.
If / . ^^ = 0, ^ is called a nullilat.
10. Theorem. Let
and
Then, we have
I . E,E,= \
{i=l....r)
k«r'=l
Inhere C'y is the minor of Cj, in \c
Further
a-
la, a, E,
If /'. refers to the E coordinates just as /to those of the e's,
r
since 2 Cy C,/, = or 1 as k :^j or /.• =j, and Jcy |- = 1.
i= 1
Hence / is invariant under a change to a new orthogonal basis.
11. Definition. Let the expression A . a^ .... a,„_i Afi^. . • ./3,„ represent
the determinant
/3l /?2 ^3 /?.
/a,„ _ 1 /?i Ta,„ _ 1 /?o /a„, _ i /^j .... /x,„ _ j /3„
In particular
^i . Uj J-p'i /Sa = /?j /a, /?2 — iSg /tti /?!
i . aj ^a, ^,(^1 ((^^3 = l^«i A, ^asi^s! = — /• ao^Uj ^/Jj/^a
= /. {SiA^.y -4aia3
These expressions vanish if ai . . . . a„,_i are connected by any linear rela-
tion, or /!?i .... /3,„ by any linear relation, or if any a is orthogonal to all of
the /?'s. If any (3, say ^^, is orthogonal to all the a's,
/ . tti Aa., ■ ■ ■ ■ a,„ A^i .... /3„, =
and
Aai a,„ _ 1 Al3i 3„ = iSi /ui Aa., a„ _i AjSo ■ ■ ■ (3„,
12
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
12. Theorem. A . aA^y + A . ^Aya + A . yAa^ =
/ . aA^Ayh + I . (SAyAab + I . yAaAi3h =
j3I aa = a/a/3 — AaAa^
A . aia2A(3i(3.,(3a = ^iI.a,AaJ(32l33 — (S,J.a^Aa2A(3i(i-i + t^3l.aiAa,A(3i^-.
= Aa, A13, 13, la., ^3—^ ai^/3i/?3 ^^a^A + A a^A/SoJ^ la^^y
I. ayAao as A(3i /J, /?3 = — ^ ' 0.2 -^Ci «3 -^/^i /^2 /^3 = • • • •
= I. l3iAl32 ^3 Aaiasa3= — I. /?„ l/3i/?3 iajaoag = ...
y I . aA(3Aa^ = al . aA^Ay^ + ^I. aA^Aay + Aa^Aa^y
13. Theorem. In general
A.ai- . ■ .a„i_i-4/?i. .../?„ = 2 . (SiIa^Aa,- ■ ■ .a,^^iA(3o. • ■ •/?„,
= 2 . Aa, A^if^z /ttg Aas . . . . a„_i A^a ..../?„
= 2 . AaiaoA^ilB^Sal.asAai. ■ ■ .a„_i A/Si- . . .(3^
(31. aiAao- ■ ■ ■a„_i-4ai. . • .a„_i = 2 .ai/ai^ag- • • -ar^^iAfia,- ■ ■ -an..!
Signs of terms follow rule for Laplace's expansion of a determinant. Develop-
ments for J-ttj A(3iy and higher forms are easily found.
14. Theorem. If the notation be used
^1^12 ••*j = ^ .:^,A.AX,....A. A?.,A(io^,.fi,_
I i"2 • 1«1
then
i«0 i«l /"3 /"3 ^.-1 f's
/X3//0 /Jl3i>'3 I^3f^s-1 ^^3(^B
I\ ."0
-^^s^«
It follows that
A . X1X2 A (1q jUi |M2
= ^fj»)-/.X,„.^{j;}
DEFINITIONS 1 .'J
Omitting X and fi
f ] 23 ) ( 2'>1 f ■■'3 )
The forms J. . . . . yl . . . . may ail be developed in this manner.
The form AV^l^ V I , where i, i„, Ji /„ are two sets of n
subscripts each chosen from among the r numbers 1 . . . . r, may be looked
upon as determining a substitution of n cycles on the r numbers, the
multipliers J'/.j^^^fii^^^^,- ■ ■ -I'/.j^Hi^ furnishing the other r — n numbers, that is,
the whole term determines the substitution
J *)i + l ■ ■ ■ • trj1'\ ■ ■ • • K I
(^ri+1 • • • JryJl ■ ■ • ■ Jn)
which must contain just n cycles. It is also to be noticed that ii^jt,
t^ 1 . ... 71. The terms in the expansion of J. . Xj . . . ./.^ -^/«o,«i ■ ■ ■ ■ f^r are
then the r! terms corresponding to the r! substitutions of the symmetric group
of order r\. The sign of each term is positive or negative according as the
number of factors / in front of the A ] \ is even or odd. Certain theorems
are obvious consequences but need not be detailed,
15. Definition. Let §(a/3) be any expression linear and homogeneous in
the coordinates of a and (3.
Also let
be formed. This is called the Q-th bilinear ^.^
16. Theorem. If e,' is any other orthogonal system,
Q.^^ = :iQ.cjel.Iejet He,
^ y • ^i 6«
Hence Q . ^^ is independent of the orthogonal system.
It follows at once that
I . ^A?.i A^fii = (r — 1 ) /Xi [li A . p.i A^fii fi, = — (;•— 2) ^;Li Au, ^
/ . ^A2.i . . . . X, A^u^ . . . . Uj, = (r — s) /. Jlj Ax, . ■ . ■ '/.^ Aui ...._«,
J. . ^Xj ?i, A^f^^ . . . j«,+i = — (r — s — 1 ) .4X1 /., Au^ ju,+i
Q . ^^ may also be written Q . y^ by extending the definition of v> ^he
coordinates of ^ being x^ . . ■ ■ x,, that is, V = 2 C; -^— .
' M'AlLAT 1.
14 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
17. Theorem. By putting subscripts on the zeta-pairs we may use several.
Thus
A.^,^,Ap^,^C2= (r-2)(r-l)p
/ . ^1 A^, ;ii A(, ^3 ^,1= ir~2) [r - 1) I A, ^i,
A^, ^2 ;ii A^i ^2 ^1 ^u = (r - 3) {>■ — 2) 4X, Afi, fi^
In general
I^, A^,...^,A^,...^, = r{r-l)...ir-s + 1)
7^1 ^^o. . . g ;ij . . . X, A^, ...^,fi,...^i,=
(r_,s) (r-s- i)...(r-s — <+ 1) /. ^1 A?.,...?., A«i • • • i"»
If s + < > r this vanishes ; i£ s -\- t = r, yve have
7 . /^j ^/lo . . . /Ij Af.li ■ • ■ f's ^
^^1 . . . ^j Xj . . . ?-s_i .4{;i . . . ^'^ j(fi . . . fi, =
(-1)' (r - 6-) . . . (r _ 5 — < + 1) .4^) . . . ;i,„, ^,«i ...//,
All . . . ?u,_i Afii . . . fi,—
18. Theorem. If 7cc, p = i z= i . . . . m — 1, then
p= ^ . «!• . . -a^.i J/;?j /3„
where /3j, (/==!.... m) is arbitrary. For, if we take the case where
m — 1 = 3, we have for /3], /J^, /?a all arbitrary, the identity
J«, ao ttj ^4/3i (i.. ^ip = /?! /ai -la, aj -4/33 /^s P — /^3 -^"i -^'^'s «3 -^/^i /^3 P
+ /^a 7/1 ^a. ag J/^j /^o p — p/aj Aa^ "3 ^/3i ^^ /^g
Hence
p 7«i yiaa a-j A(3i ^^ ^3 = (3^ 7«, Aa-, a^ Afi.^ /?3 P — /^g -^"1 ^aa "s ^i^i /?3 P
+ /^g 7ai -(4rx3 Kg J./3] Z?, p — -4a] ao a^ A(3i ^., /^g p
Since 7a) p == /'/.2P ^^ -^"sP ^ i therefore identically
7a] ,/?] Txi ^a^ag A^.^fS^p — Ia^[3.,IaiAaoa:tA^i i5gp + lixS^Ia.iA'XMj -^/^i/J^p =
with two similar equations fur a.,, Wg. Therefore, since /3], /^o, .<i?3 are arbitrary
7ai Aa^. ttg ^/3i /^., p = 7a] Ja^ ag ^1/3] /i., p = /a, ylaj ag Aj^.^ /3g p —
or else, for any /3j, /?3, /?g,
/aj Aa.^ ag -4/3] /^o /?3 = ^
DEFINITIONS 15
This is impossil)le, lience
p /ui Aa.^ (la A(3i (3.. /?3 = — via, a.. «;, AfSi (I, /?« p
or p = -4«i a., «3 -4/:?j /?3 /iy [ii
where /?,, /?a, /?;,, ^^ are arbitrary. A similar proof holds for the general case.
This ciilciiliis would enable us to produce a tlieory of all bilinear functions
(2(ap), and thus the so-called algebras.*
19. Definition. A subregion^ consists of all hypercomplex numbers which
can be expressed in the form a = «i p, -f- <';• P;; + • • • • + "iPi wherein p,, p^,
...., Pi are given, linearly independent, numbers of the range of the algebra.
20. Theorem. An unlimited number of groups of m inilependent numbers
can be found in a region of m dimensions.'^ Any group is said to define
the region.
21. Definition. Tlie calculus of these entities is called an uhjehrd, if it
contains, besides addition, another kind of combination of its elements, called
multiplication. The algebra is said to beef finite dimensions, when it depends
on r units, r being a finite number. Of late the term finite has been applied
to algebras the range of whose coordinates consists of a finite number of
elements.
Multiplication is usually indicated by writing the niuiibers side b}' side,
thus, a^ or a. /3. Upon the definition of multiplication depends the whole
character of the algebra.* The definition usually given is contained in the
statements :
if
a = 2 «; Cj
k=l
then a. ^ ^ y
if
l....r
c,= 1 . or, . hj .
Yijk
{Jc=l,2....r)
The constants j/;^^. are called constants of muUipUcation. If tnultiplication
is defined in this manner the algebra is called linear. The products o, . hj are
defined for, and belong to, the range of coordinates. The constants of mul-
tiplication also belong to the range, and their products into a, t, are defined
for, and belong to the range. Algebras whose constants are such that
|yJ^^. = y^j/. are called reciprocal. If y,'j^= yfj^, they ViVQ parastropldc.
22. Theorem. If multiplication is defined as in § 21, then
a . (/? + y) = a . /3 + a . 7 {a -\- ^) . y = (x. y + ^ . y
{a^- ^).{y + h) = a.y ^a.h-V ^.y + [i.h
This is usually called the distributive law of multiplication and addition. An
algebra may be linear without being distributive.^
'Whitbheab 1, p. 123. 2Cf. Whitehead 1, p. 123. ^cf. Gibbs 2, Macfarlase 4, Shaw 1.
*Shaw 9. 6 Dickson 7.
1 6 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
23. Definitions. In the product a/?, a is called the facient,^ or the le/t
factor, or the pref actor f /? is called the faciend,^ or right factor, or post/actor.'^
The latter names will be used in this memoir.
If there is a number a^ in the algebra, such that for every number of the
algebra, a, a ag = a = a^a, then ao is called the modulus^ of the algebra.
If we have a ao = a aja ^^ a we may call a^ a post-modulus.
If we have Oq a = a a .ao :|: a we may call Oo a pre-modulus.
In defining an algebra, the existence of a modulus may or may not be
assumed. When for all numbers a,/?, we have a ^ = 13 a, the algebra is
called commutative.
When for any three numbers a, /3, y we have a .{^ .y) = {a . (i) . y,
the algebra is called associative.*
24. Theorem. If an algebra is linear, the product of any two numbers
is known when the products of all the units are known. These products
constitute the midliplicafion tabic of the aigebra.
25. Theorem. In an associative algebra the constants of multiplication
satisfy the law
r r
2 ra« y,jt = 2 y;,js y^t {i, 1^,j, t = 1 r)
s=l s=l
26. Definitions. If a . a ^ a" = a, then a is called idempotent.
If a'" = 0, m a positive integer, then a is called niljwtent, of order^ wj — 1.
If a, 6' = 0, then a \s p)re-nilfactorial to /3, which is post-nilfacturial to a.
Ifa/i = = /^a, then a is nilfactorial to |3, and (i to a.
27. Definition. The expression I. a (3 is sometimes called the inner or
direct pjroduct^ of a, /3 and written a* /3. Further, the expression
Q{a^) = Y a,bj .e,Iej{)
is called the dyadic of a (5, and written a /?. It is thus an operator and not a
product at all. The use of the term product in similar senses is quite common
in the vector-analysis, but it would seem that it ought to be restricted to
products which are of the same nature as the factors. Gibbs, however, insisted
that any combination which was distributive over the coordinates of the factors
was a product.'
There is no real difference between the theories of di/adics, matrices, linear
vector operators J bilinear forms, and linear homogeneous substitutions, so far as the
abstract theory is concerned and without regard to the operand." If we
' Hamilton 1, B. Peirce 3. 'Taiieii 5.
^ScuEFFKKS 1, Study 1, who calls it one (Eins), identifying it with scalar unity. Some call it Ilaupt.
einbeit. Cf. Shaw 1.
<B. Peihce3. 'B. FeikceS. « Gibbs 3. ' Giuns 3.
*Fkobenius 1, and any bibliography of matrices, bilinear forms, or linear homogeneous Bubstitutlons.
Cf. Laurent ], 2, 3, 4. See Chap. XXX this memoir.
DEFINITIONS 17
denote the operator Q (a/3) by 4>, then the bilinear form Scy x, yj may be written
J.otpa or I.a^p, where <^ (or ^') is called the conjugate, the trnnsvertie, or the
transpose of 4). Besides the ordinary combination of these operators by
"multiplication" Stki'IIANos* defines two other modes of composition which
may be indicated as follows in the notation developed above :
(1) Bidlternafc composition in which
<^i . <^2 is equivalent to „ , C\., Tp'Ap" A<pi a' ^o a"
<?>i • <?>2 ••■•<?>« is equivalent to -^| 6j , I^'Ap" .... p'"* A(pi a' .... <^, a"
C,. , indicates that the sum is to be taken over all terms produced by permut-
ing in every way the subscripts on the ^'s.
(2) Conjunction, which corresj)onds to the multiplication of algebras,
and is equivalent to taking <^i and 4)3 on different independent grounds
ei .... e^, e[ ... . e^,, whose products Cje,' define a new ground
fiy = Cjc; {i=l .. .. r,j = I .... /)
Thus <^i X (?)a = ' '2 ' ' 2 'eg' cf} ea, Teji
i.j k. I
2. definitions by independent postulates.
28. Definition. Three definitions by postulates proved to be independent
have been given by Dickson." The latest definition is as follows:
A set of 7j ordered marks a^ .... a^ o? F (a field) will be called an n-(nple
element a. The symbol « =: (aj .... a,.) employed is purely positional, with-
out functional connotation. Its definition implies that a =■ h if and only if
ttj ^ 6, .... a,. = b,. .
A system of «-tuple elements a in connection with n^ fixed marks yij^
of i*' will be called a closed system if the following five postulates hold.
Postulate I: If a and h are any two elements of the system, then
« = (oj + 6, .... a, + h^) is an element of the system.
Definition : Addition of elements is defined by a ® 6 = s.
Postulate II: The element = (0 .... 0) occurs in the system.
Postulate III: If occurs, then to any element a of the system corre-
sponds an element a' of the system, such that aea' = 0.
Theorem : The system is a commutative group under 9.
Postulate IV: If a and h are any two elements of the system, then
P = {jh • • • • 2\) is an element of the system, where
l..r
Pi = 2 ajb^ yj„i (j = 1 r)
Definition: Multiplication of elements is defined by a ® 6 =^.
' Stephanos 6. 'Dickson 5, 8.
18 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
Postulate V: The fixed marks y satisfy the relations
r r
2 r^tj yji-i = 2 y, ij y,ji (s, t,k,i= 1 r)
J=l 3=1
Theorem: Multiplication is associative and distributive.
Postulate VI: IfTj .... T,. are marks ofii^such thatTjrtj + • • ■ +T,.flr:=
for every element (oj .... o,) of the system, then t^ = . . . . t^ = 0. [This
postulate makes the system r dimensional'^.
Theorem : The system contains r elements s^ = {(l^■^ .... a.-r), i = 1 . . . . r
such that I fly I -^ 0.
Theorem: Every ?--dimensional system is a complex number system.
Generalization: If the marks a^ . . . . a,^ belong to a field F^; and if
flr, + 1 • • • • (^r,+r. beloug to a field F^; . . . . , if a corresponding change is made
in postulate VI ; if further yj,,i = 0, when j, k, i, belong to different sets of
subscripts, then we have a clcsed system not belonging to a field F}
3. DEFINITIONS IN TERMS OF LOGICAL CONSTANTS.
29. This definition is recent, and due to Bertrand Russell. By logical
constants is meant such terms as class, relation, transitive relation, asymmetric
relation, ichole and ^)a?-<, etc. Complex numbers are defined in connection
witli dimensions, or the study of geometry . The definition in its successive
parts runs as follows : "
30. Definition. By real number is meant any integer, rational fraction,
or irrational number, defined by a sequence. These have been discussed
previousl}', in the work referred to.
A hypercomplex number is an aggregate of r one-many relations, the
series of real numbers being correlated with the first r integers. Thus, to the
r integers we correlate flj, o, .... a,., all in the range of real numbers. This
correlation is expressed by the form
The order of writing the terms may or may not be essential to the definition.
The e indicates the correlation, thus Cj is not a unit, but a mere symbol, the
unit being le^. The remaining definitions, addition, multiplication, etc. may
be easily introduced on this basis.
Theorem : Hypercomplex numbers may be arranged in an r-dimensional
series.
31. A like logical definition may be given when the elements belong to
any other range than that of "real" numbers.
4. ALGEBRAIC DEFINITION.
32. The preceding definitions are of entities essentially multiplex in
character. The units either directly or implicitly are in evidence from the
'Cf. Cakstens 1. «B. Russell 1, pp. 378-379.
DEFINITIONS 19
beginning. It seems desirable to avoid this multiplicity idea, or implication,
until tbe development itself forces it upon ns. Historically this is what hap-
pened in Quaternions. Originally (|uaternions were operators and their
expressihility in terms of any independont four of tlieir nnniber was a matter
of deduction, while Hamim'oi^ always resisted the coordinate view. The fol-
lowing may be called llic algebraic definition, since it f(dl()\VK the lines of
certain algebraic developments.
33. Definition. Let there be an a.ssemblage of entities ^(, either finite or
transfinite, enumerable or non-enumerable. They are however well-delined,
that is, distinguishable from one iinothcr. Further, let these entities be subject
to processes of deduction or inference, such that from two entities, A^, Aj, we
deduce by one of these processes, passing from -4j to Aj, the entity A,/, which
we will indicate by the expression
Ai OAj = A/, {Ai, Aj any elements of the assemblage)
A different process 0' would generally lead to a different entity A',,; thus
A, a A; — Al
(These processes may be, for example, addition °d, and multiplication 0). It is
assumed that these processes and their combinations are fully defined by
whatever postulates are necessary. Then the entities A, and the processes
O, 0' . . ■ ■ are said to form a calculus, and the assemblage of entities will be
called a range.
34. Definition. Let there be given a range and its calculus, and let us
suppose the totality of expressions of the calculus are at hand. In certain of
the.se, i^/l, M.,. . . .M^, let us suppose the constituent entities Ai, Aj . . . . are
held as fixed, and that we reduce the totality of expressions modulo these
expressions M\ that is, wherever these expressions occur in an}- other expression,
they are cancelled out. Then the calculus so taken modulo M is called an
algebra.
For example, let the range A be all rational numbers. Let the expres-
sions iV be r • . -, ^
Then an expression like 4 — 8 may be written 4i -|- 4 + 4 — 8 = 4i; an
expression like x~ -f 9 becomes ar + 9 — (9 -f 9^") = x- — 9^- ; which may be
factored into {x + sy) (a; — Zj) or (x + 3^) (x + Sy ).
In this manner we have a calculus in which will always appear the
elements i, j {or J andy- as we might find by reductions). Modulo i + 1 and
y"-f 1, certain expressions become reducible, that is factorable, which other-
wise cannot be factored. We call the expressions xi, xj, xj^, in this case,
where x is any rational number, negative numbers, imaginary numbers, and
negative imaginary numbers. We consider i andy as qualitative units, although
perhajDS modular units would be a better terra.
20 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
35. It is not assumed necessarily that there is but one entity A^ for any
given expression, for we may have two expressions alike except as to the
elements that enter them. Thus we might have
"^e'
31
l;'::}
36. Definition. In any case we shall call the expressions 31 the defining
expressions of the algebra, and the elements A^ (such as i,j) entering them the
fundamental qualitative units.
37. Postulates :
I. It is assumed that the processes of the calculus are associative.
II. It is assumed that the processes which shall furnish the defining
expressions shall be those called addition e, and midtiplicatiun ®.
III. It is assumed that the process ®, multiplication, is distributive as to the
process ®, addition. Tiiat is
Ai ® {Aj ® A^) = {A, ® 4) © {A^ ® A)
{Aj © A^) ® A, = {Aj © A-) © (A^. ® Ai)
38. The comrautativity of multiplication is not assumed. Further, the
general question of processes and their relations is discussed, so far as it bears
on these topics, in XIII, hence will not be detailed here.
It is evident according to this definition that an algebra may spring from
an algebra. Hence the term is a relative one, and indeed we may call a cal-
culus an algebra if we consider that the calculus is really taken modulo
A,OA^ — A, A,0'Aj-M, etc.
That is, the equalities or substitutions allowed in the calculus make it an
algebra. The only calculus in fact there is, is the calculus of all entities
J.,, Aj, Ak, etc., which permits no combinations, that is, no proces.«es, at all.
From J.;, Aj, .... we infer or derive nothing at all, not even zero. The calculus
of symbolic logic is thus properly an algebra.
Any definition of an algebra must reduce to this definition ultimately,
for the multiplication-table itself is a set of r" defining expressions. That is,
we work modulo^
r
Cj gj — 2 Yijk ek (*, y = 1 • • • • r)
k=l
39. Definition. If the range of an algebra can be separated into r sub-
ranges, each of which is a sub-group under the process of addition e; so that
an entity which is the sum of elements from each of the sub-ranges is not
reducible to any entity which is a sum of elements from some only of the
sub-ranges; then the algebra is said to be (additively) r-dimensional.
iCf. Kboneckeu 1, where this view is very clearly the basis for commutative systems.
DEFINITIONS 21
40. It iH to bo noted tlwit an algebra may be /■-dimensional and yet have
in it /• + A' distinct qualitative units. ThuH, ordinary positive and negative
numbers form an algebra of two units but of only one dimension. Ordinary
complex numbers contain four qualitative units, but form an algebra of two
dimensions.
The defining exprcs.sions determine the question of dimensionality. For
example, let the defining expressions be
f e'i— I e'i — 1 e^Ci — e\ e.^
t ei + e\ + 1
whence we may add
^1 *2 + ^ ^2 + <^2 ^3 *1 ^2 ^1 1 ^1 ^2 ^1 ^2 1 Ci 6^ ^ 6^ Ci — 1
We have here two more defining expressions than are needed to define an
algebra of six units, hence the algebra becomes four-dimensional. The
problem of how many defining expressions are necessary to define an algebra
of r units has never been generally solved even for such simple algebras as
abstract groups. If the algebra is finite of order r, a maximum value for the
number is 'r. But a single expression may define an infinite algebra.
Nothing, so far as known to the writer, has been done towards the study
of these algebras of deficient dimensionality.
n. THE CHARACTERISTIC EQUATION OF A NUMBER.
41. Theorem. Any number ^ in a finite linear associative algebra which
contains a modulus, e,,, and whose coordinates range over all scalars, satisfies
identically an equation of the form A' (^) = 0, and equally an equation of the
form A" {^) = 0. In each case. A' (^) or A" (^) is a polynomial in ^ of order r,
the order of the algebra.'
The function A'. ^, called the pre-latent function" of ^, has the form
2 . Xi y.ii eo — (; 2 . Xf yi2i 2 • x,- >/,■,,
The function A". ^, called the post-latent functionr of ^, has the form
2.x,yii3 'S,.XiYov>€Q — ^ 2.x,.y,.(2
A" . ^ =
S-a-jyiir S.Xjysir 2 .», y^ir «b— <f
'The relation between this equation and the corresponding equation for matrices is so close that we
may include in one set references to both; Cayi.ey 3; LaguerreI; B. Peirce 1,S; Frobenivs 1,2;
Sylvester 1,3,3; Bucbbeim 3; Scuefpers 1,-', 3; Weyr 1,5,8; Taher 1,4; Pascu 1 ; MoLlEN 1;
Cartan 2; Shaw 4.
^Cf. Taber 1.
2
^2
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
In each case 2 stands for 2 . These functions may be expanded according to
1=1
powers of ^, taking the forms
A'.^ =^^—m[ ^^-' + mL T"" + {—Ymle,
A".^ = r - m[' r-' + mi,' ^'-'^ + (-)'• ml' e„
In certain cases (viz., when the algebra is equivalent to its reciprocal) these
two become identical. (The absence of a modulus does not add to the
generality of the treatment.) These equations exist for all ranges of
coordinates.
42. Definition. The coefficients m[ and wjj' are respectively the pre-scaJar
and the post-scalar^ of ^, multiplied by r; that is, if we designate the scalar
of ^ by S .^, we have
g,y_.m[ ^„ X _ »<
r r
If we indicate S.^^ by Si, we have by well-known relations from the theory
of algebraic equations
rSi 1
rSo rSi 2
rSs rSs rSi 3
m: =
Theorem: The symbol S obej's the laws^
i—l
(a, any scalar) *$" . Co = 1
43. Definition. The number F'. ^ = ^ — S'. ^ is the pre-vector'^ of ^,
and the number V". ^ = ^ — /S'". ^ is the post-vector^ of ^. By substituting
these for ^ in the identity for nij in §42 we arrive at various interesting and
useful formulae.
44. Definition. If the two equations A'. ^ = 0, A". ^ = are not identical,
the process of finding the highest common factor will lead to a new expression
A . ^ which must vanish. When the two equations A'. ^ = 0, A". ^ = are
identical we may also have ^ satisfying an equation of lower order; let the
lowest such be
A.^ =
This single equation is called the characteristic equation of ^, and A . ^ is the
characteristic function of ^.* [The pre-latent equation was called the identical
equation by Cayley, characteristic by Frobenius and Molien, and this lower
■Tabek 2, 3, 4, G. Cf. Frobenius 14, §4. Fuouenius called m, the Spur of ^.
»Tabbii2, 8, 4, 5. » Cf . Taukh 2, 8. «See refereuces to §41.
THE CHARACTERISTIC EQUATION OK A NUMBER 23
equation has been called " limu/fj/eicJinnf/" hy Molikn, " GnimJfjkicJturifj" by
Wfvk, idnitlcal eqwition and fuiUhimndal eqitatioii by Tahku, charadtrihiic
equation by SciiEFFKUS, and in some cases it is the reduced churucleristic
equation.]
45. Theorem. The characteristic function is a factor of the two latent
functions.'
46. Definition. The order of the characteristic function being r';!?-, it niny
be written
i^-giCoY' {^—9,.^oYp 1^1 + + ^. = ^'
The scalars 7, .... </,, are the p distinct latent roots of '(. The exponents
Hi .... |Up are the j^ suh-mulliplicitiea of the roots of ^. The factor ^ — ^, Co 'b
the latent factor" of the root g^.
Weierstrass called (^ — i/i Co)> •''•"^ powers, elementary factors {eltmen-
tartheiler), particularly the powers: A;^, { /c^j _i,i • • • • /«i, i- See MuTU 1 for
references to this subject, or Weierstrass 1 ; Kronecker 2, 3, 4; Frobenius,
Grelle 86, 88 ; Berliner Sitz-her. 1890, 1894, 1896.
47. Theorem. For a fixed integer i (1^ i ^ i)), there is at least one solution,
a, {a -^ 0) for each of the equations
A solution of the k-th equation is a solution of those that follow. If (T^t
is a solution of the 7i;-th of these equations, then among the solutions of the
k+ 1-th equation, which include the solutions of the previous equations, some
are linearly independent of the entire set of solutions ct,;,j of the k-th.
equation.^
Theorem : The solutions of these equations for different values of i are
linearly independent of each other.*
48. Definition, The number
y _ (? — ffi e p)"' • • • • (^ — 9i-i eoYi-i (? — 9i+i epYi+i . ■ ■ • (^ - gr^ e^Yl
'~l9i~9iY' •■■ {9i-gt-iY^-' {9i—9i+iY^^' ■■■■{9i—9pY^^
is the i-th latent of ^; it corresponds to the root g^. There are thus p latents of ^.
49. Theorems. The product of Zi and any number of the algebra is either
zero or else it is a number in the region of solutions of the equations in §47.^
We may symbolize this by writing Zi\a\ = {^i\ The region ]^, [ is called the
i-th pre-latent region of ^. There are correspondingly post-latent regions of ^.
' Taheu 1 ; \Vevr8; Molien 1 ; Frobenius 14. * Taber 1 ; Whitehead 1.
^TabkrI; Whitehead 1 ; Cartas 3. *Tabee1; Whitehead!; Shaw 4. » Suaw 4.
24 SYNOPSIS OP LINEAR ASSOCIATIVE ALGEBRA
The p latent regions of ^ together constitute the whole domain of the
algebra.' It is obvious that the Z's are such that if i ^^i,
Z,Zj=0 {Z,-e,YiZ^=Q
50. Theorem." The p pre-(post-) latent regions are linearly independent,
that is, mutually exclusive, and together define the ground of the algebra.
Each latent factor annuls its own latent region but does not annul any part
of any other latent region. The i-th pre-latent region may not contain the
same numbers as the i-th post-latent region. The dimensions of the i-ih. pre-
latent region are given by the exponent of the i-ih. latent factor as it appears
in the pre-latent equation. The pre-latent equation contains as factors only
the latent factors to multiplicities jU-, such that
i^iliii {i=l .... p)
V
1=1
Likewise the post-latent equation contains as factors only the latent factors to
multiplicities ^[' , such that
ill = («i (i = 1 V)
i=l
51. Theorem.' The pre- (post)- latent region ]^jf contains jU; sub-latent
regions \I,a\, l^a'h • • • •, ]2,v.,[, where each sub-latent region includes those
of lower order, say {Xik] includes ^2^'} if k' <^k.
The region \'S.ik\ is such that (^ — gieof ^2«} = 0, but in \Xik\ is at least
one number ct,,, for which (^ — Jj'ieo)*"^ ^a- ^ 0.
52. Definition. For brevity let ^ — gi 6^ = 61; then, in ]^,j-, $^''1 annuls
certain independent numbers which no lower power of d^ annuls. Let these
be Wij in number, represented by
sn ^31 ^Wiii
Of course any lo^^ independent numbers linearly expressible in terms of these
would answer as well to define this region, so that only the region is unique.
Then each of these multiplied by 0, gives a new set of ivn numbers independent
of each other and of the first set. Let these be
In general we shall have for the products by powers of 6j a set of numbers
linearly independent of each other,
7t = . . . . |«, — 1
0" ^yi = cyi r* =
W
It
'Taber 1 ; Whitehead 1 ; Suaw 4.
'TabekI; WuiTEiiEAUl; Shaw 4; WeyrB; Buciiheim 3, 7, ".».
'See preceding references.
THE CHARACTERISTIC EQUATION OF A NUMBER 25
The region made up of, or defined by, these numbers will be called t\ie Jirst
prc-sltear rcfjioH^ of the i-th latent region. It tnay he represented by ^A'J^'f.
Let there be chosen now out of the numbers remaining in the i-th latent region,
w.,j linearly independent numbers which are annulled by that next lower power
of 6i, say jti,.,, which annuls these w.^, numbers, but such that 6'"'-"' does not
annul them and such that Sf's '^ does not annul any number which 0f<a does
not also annul. These numbers and their products by powers of 6i give rise
to the second pre-sJiear region, {(ii-^ <[ (li)
We proceed thus, separating the i-iU latent region into C( shear regions,
|Xi',.[, , \X^^, containing respectively (iii = fx^) ^^ >'ht, , i^ic, w^a
linearly independent numbers, with
Cl
j=l
There is a corresponding definition for the post-regions.
53. Theorem.'- The pre- and the post-latent equations are (using accents
as before to distinguish the two sets of numbers)
n di^^'n'-'v = j=\ .... c[
< = i
YiBT"n''"ii=0 j=\ .... c;'
54. Theorem. If all the roots gi vanish, ^ is a nilpotent, and for some
power ^ we have ^'" = 0.
Further, for every number there are exponents (Xj, ^", such that
If ^ and a are of the same character,^ (aa) then for any power /«<,, ^"'^ a
and (T '('"I- are nilpotent.
The product may not be nilpotent if ^ is of character (a/3) and a of
character (/3a). If the product is not nilpotent the algebra contains at least
one quadrate. If an algebra contains no quadrates, ^''^ a and (T^"* are nilpotent
for all values* of a and fi^.
55. Definitions. When the coefficients in the pre-latent (post-latent)
equation vanish in part so that
then ^ is said to have racuifi/^ of order j!^^'. There are «o zero-roots, and one
or more solutions of the equations
^0 = ^-CT = ^"'o a = (io = fil)
'SiiAw4 -Su\w4. 2See§5(i. 'Cartan 2; Taber 4. > Sylvester 1 ; Taber 1.
26 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
The solutions of ^ff = define the mdl-region of ^. The number of inde-
pendent numbers in this region (its dimensions) is \\\q first mtJlity of ^, say A;j.
The h^ independent solutions of ^^a = 0, ^g if 0, define the first sub-null-
region of ^, of second nullity h^; proceeding thus we have^
The vacuity of course is given by the equation
^0 = ^1 + + ^^^0
The characteristic equation, it must be remembered, contains ^'^'' as a factor;
the pre-latent equation ^'^''', the post-latent i^'^"". The partitions of /.ig which
satisfy the inequalities above give all the possible ways in which the sub-null-
regions can occur.
56. Theorem.^ Each latent factor, ^,-, is a number whose pre-latent (post-
latent) equation will contain ^f', and whose characteristic equation will contain
^f . The nullities of ^j are given by the equations
Ki-n = «'i^ + <«'3i < = Oor 1
hi = ^li + *^2i + + ^1 w^c,- u + 's «fc<i h, «3 = or 1
hi = W^i -\- W^i + + Wc,i
The vacuity fil = /"a w'u + /^,;3 ^'2; + + f-i-iu'^ui
57. Theorem. The number ^ may be written^
wherein the numbers xj 4), (t = 1 /)) satisfy the following laws :
xf = Xi Xi xj =0 if i :^ j
Xi%=zQ =^jXi ii*+y
^i ^i = ^i = <?». ^i
The numbers Xi and ^^ are all linearly independent and belong to the
algebra, at least if we have coordinates ranging over the general scalar field.
58. Theorem.* Let ha ^^ + + /',>.. 1 <?>r' ' = ^o then if F (x)
is any analytic function of cc, F' (x) its derivatives,
jr,'^=i^S^F{g^).x,-\-F'{gd.%+ ^^^^^ + |~ /)f *^''"'}
'8rLVESTER2; Tabku 1 ; BuouueimS; Whitehead 1.
!§52: 3 Study G; Shaw 7. <Siiaw7. ('f. Taiiek 1 ; Svlvustek 8.
THE CHARACTERISTIC EQUATION OF A NUMBER 27
59. Theorem. Tlie (liirerent number.s of the algebra will yield a set of
idempoteut expressions e, .... c^, such that if i ij:/, i^ j = i .... a
^'/ = ei efej=0 — CjCi gg = e, + -\. e^
and hence the numbers of the algebra may be divided into classes \Z^p\, such
thiit if ^,.3 is in the cla.ss \Z^^\, then
The subscripts a, /? are the diameters^ (pre- and post- resp.) of ^„^. In this
and similar expressions ^j.y = when x -^ 1/, S'^,, = 1 when x =y.
60. Theorem. The product of lif^^ and ^^j is given (when it does not vanish
on account of properties not dependent on the characters) by the equation^
The numbers ^„„ form a sub-algebra, (a = 1, . . . ., a).
61. Theorem. Let the characteristic equation of ^ have q — 1 distinct roots
which are not zero, and let v — 1 be the lowest power of ^ in this equation.
Tlien if
we have''
i=l
<J-1 q-1
62. Theorem.* If e^ rf: ^ x^, then e„= 2 Xi + x,,, where x, belongs to
i=l i=l
the root zero and
Theorem : It also follows, that, if
^^=l<-(|E^yr'
then i'', ^ =: ;Cj
63. The use of the two sets of idempotents of ^, the pre- and the post-,
enables us to find partial moduli, which are not necessarily invariant, and the
modulus, which is invariant.
For example, let us have the algebra
ei
eg
^3
«!
ei
e^
^2
es
ea
«8
Then €„= e^ -\- e^
'SiHEFPEKS 1, 3, 3; Caktan 3; Hawkks 1; SuAW 4. Cf. B. Peikce 1, 3. Fkobemus 14.
-■ See references to § 59. 3 Taber 4. « Tabbb 4.
28 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
If we put ^ = ej + Cg, we find
^ {ei + e3) = {e, + e,) ? - e„ = ^ ^3 =
hence the characteristic equation (^ — Cq) ^ = 0, and by §§61, 62,
xi = ^ X2 = e2 — e-i eo= xi + x„
These determine the same algebra (in the sense of invariant equivalence)
Xi
Xs
^3
^I
Xi
J£2
Xo
^3
^3
«3
and the partial moduli are not the same as before, being e^, e^ in one case,
Cj -1- eg, Co — Cg in the other.
64. Theorem. If ^,- is any number in the i-th pre- (post-) region of ^, and
if a is any number of the algebra, then ^j cr (cr ^,) is a number wholly in the
i-th pre- (post-) region.' Consequently the numbers in the i-th pre- (post-)
region form a sub-algebra.
65. Theorem. Let the numbers defining the i-th post-latent region of ^ be
^ii, where
y = 1 Ci 5=1 Wji f= I [^^J
We have of course
S8f • ^i — Cst + l
so that
Then by § 64 the product of any number a gives
" • Ssl •^ • "-Mt^ hUV
Hence when these coefficients a are known we know the product of a into
any number of the form ^H, for
where" Euv^i^i must be zero if v + f — 1 >/u,a.
66. Theorem. If r is any number of the algebra which satisfies the equation
T. 6/ = 0, where r . 6,'^ ' 4^ 0; then r must be in the region (§ 13) 2'^, and
in no lower region.''
67. Theorem. If r is any number of the algebra, and if cr,^. lies in the
region i'/., hut in no lower region, then tct,, lies at most in the region 2'/g,
and may lie wholly in lower regions.*
I SuAW 4. «8haw4. »Siiaw 4. ■>SiiAw4.
TllK ('IIAI{A(n'EUISTIC EQUATION (JK A NUMBER 29
68. Theorem. Let i'' ho the region to which . 0'"ti~'' reduces the whole
i-tli post-latent region, and generally i'" be the region to which . fl'^a""' reduces
the latent region. Then if t is any number of the algebra, and cr*" any number
of the region i'", then
ra" = 'a'", a number of the region 1.'" or lower regions.'
69. Theorem. If oJ| is a number common to both regions 2'* and 2,(, then
T . aJt = 'a'tl, a number in the same regions.^
70. Theorem. Let
Sl^. be the region \^i\\, (s = 1 . . . . w^)
then St = S'^..__a II belongs to the regions 2''"^-'' + ' and 2.-,„ ._„ + ,. Then if t
is any number, t . «S''i = | S't!, ■ ■ ■ • S'l,ul . . . .\ for all values of t subject to
the conditions
a«> I a 5<" < 6 a<'> + 6<" = ^,;, + 1
This may also be expressed in the following statement :
where y< flu, and CJ; belongs to S^^.^^y^^, and c^j belongs to Sl.._t + i.
Hence
2/ ^ « fiik — yt fiy — t
that is
y = t + ^ik — fiij
Or finally,^ if jm^^. < fA^j, then i«,^. i ?/ > <
if /"a>i"u, then ^a> ?/ = i +fiik — f^ij
It is to be remembered also that
It is evident that the products into £?,i determine all the other products.
71. Theorem. Since the units of the algebra may be the numbers £■**, as
these are mutually independent and r in number, it follows that among the iv^
constants of the algebra, y, which the coefficients a reduce to in this case, there
are many which vanish and many which are equal. The units may be so
chosen in any algebra that the corresponding constants y become subject to
the equations for the coefficients a in § 70 [but this choice may introduce
irrational transformations].
'Shaw 4. -^ShawM. ^Soaw 4.
30 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
72. Theorem. Since the iderapotents for ^, viz., xj, X2, • • • • Xj„ may be
used as pre-multipliers as well as post-multipliers, the units ^ij, and therefore
all units, may be separated into parts according to the products
X, . Hi (a = 1 . . . . iO
As these parts are linearly independent, and as the z-th region is defined
already by the units p^i, it follows that the independent units derived by this
pre-multiplication must also define the region, and as the shear regions were
unique, their number for each shear remains the same as before. We may
use a new notation, then, indicating the pre- as well as the post-character of ^,
and at the same time uniting y and s into a single subscript, thus the units are
^V \u't' hn"t"j ^i"
where
U"— 10^,, + . . . + Wj„ i„ + S U' = U\,, + . . . . +Wj„i„ + s
73. Theorem. Let us return to the equation in §7 0, in the new notation,
'■ • SMt — ""xy <^x?y
y-fixlyH y=t + fifix — l-hu a'/y =a'fy-l= — «it-t + i
If r is confined to expressions belonging to the region \^ll\, then letting ry
be any such number,
—^aa a y a V" j^aa a t^a l^ ^ ^ aa a Ka
^ 1 • Kul ^ "arl i^xl "T -^ "-xl Kxy
If we let
then
-Tj" . CTj" = 2 al\ zl" °^^i -f terms for which y > 1
Hence if we let tJ° be in turn each unit °^° in this region, we shall find
from al" by the process used in the beginning of the problem, certain numbers
idempotent so far as this region is concerned, and which will be linearly
expressible in terms of °^°i. These new x'?. are linearly independent and
commutable with x„, since, if x'^ is one of them, x^xl==- x'^^= x'^x^. Hence
x„ must be the sum of them. We might therefore have chosen for ^ a number
which would have had these idempotents, and we may suppose that the
number 'C, has been so chosen that no farther subdivision of the idempotents
is possible.^
74. Theorem. It is evident tliat, as the expressions in the i-th latent region
of ^ form a sul)-a]gebiii, we may choose one of them ^[ just as we clioose |,
'Cf. MoLiBN 1.
THE CHARACTERISTIC EQUATION OF A NUMBER 3 1
and using it as a post-multiplier, divide this i-th latent region itself into sub-
regions corresponding to the latent regions of ^1 in I^J. Each such sub-region
becomes a sub-algebra. We may evidently so proceed subdividing the whole
algebra into sub-regions until ultimately no sub-region contains any number
which used as pre-multipiier has more than one root for that sub-region.
This root may then be taken as zero or unity. If then the sub-region be
represented by cr, , a., . . ■ ■ a,.,, we have for every number
a = ^xj<rj T = l f/j Gj {j= I .... /)
T a =: ^ T -f- t' t/ a = (/ r' -\- r"
Hence if x, is the partial modulus for this region delined by
o' - hga"^' + ^'-^(*~ ^^ fa"-'' .... + (-)"' y" i a
^.= ^ 7
we nmst have a = g xi + ^^^ -\- other terms whose post-product by r is zero.
Multiplying every number then by x^ . () we arrive at a sub-sub-region
which gives a sul)-algebra whose modulus is x^, and such that if a is its
character, every number in it has the character
(ua)
This algebra is a Peirce algebra. Its structure will be studied later. The
Peirob algebra is the ultimate subdivision by this method of the algebra in
general and its structure really determines the main features of the structure
of the general algebra.
75. An algebra may contain an infinity of units, in which case it may
not have an equation at all. Thus the algebra may have fur units
oo
so that p = i a-j e^
!=0
It may very well happen then that pa = t<y has no solution. The theory
of such algebras will be developed in a later paper.
76. Theorem. Let the general equation of a number ^ be
Let us put ^- — ???! ^-|-(T = 0. Tiien we may eliminate ^ from these two
32
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
equations, by using* determinants, arriving at an equation in terms of a of
order r. Thus we have
1
1
+ "'s
ni,
m.
a
(-Ifw,
m.
=
or
1
— WJl
+ nio . . • •
1
— wi ....
a — m„ . • • ■
a — ?»2
OTg
— m^
O — tOo
WJg
1
— ?Hl
(T . . . .
1
— nil ....
The highest powers of this equation are
a"— 2771, CT*— ^ + =0
Hence the sum of the roots of is 2w2.
77. Theorem. In general if ^* — Wi ^*""^ + .... + ( — 1)' a^ = 0, we find
in the same manner a determinant of order r -\- s, reducing to one of order
r in (Tj, the first two terms becoming
,r_l +
Hence for any such number
we have the sura of the roots of a, equal to stti^. Hence the s-th scalar coefficient
m, of ^ is i into the scalar coeflBcient of order unity of {—Y^^^x^'^ (^) > '^^
78. Theorem. We may also find the general equations of tlie cr's, and in a
similar way of the ;^'s.
79. Theorem. In this way one may form the equations of powers of ^, or
of any polynomial in ^.
80. Theorem. Let there be formed for any number p, the products
pe, (1=1.... ;•)
The e< form the basis and are orthogonal. Then we have (p — (j) a = 0, when
V . I .e, (pp,) I . ejG = (J I . c'iG {i = l .... r)
TIIK CHAUACTKIIISTIC KtiUATK^N OK A NlIMriEH
33
Hence
/. e, ((X',) — J/ /.e, (j;e.,)
=
or
gr_fjr-i V , / . e, (pC,) + f ■' ^ I .C^ ACj ^(pC,) (pCj)
=
i- 1
lj=l
This, however, must correHpoiid to the general pre-latent equation of p, and
therefore
Wg = ^ . /. Cj ^e; -4(pe,) (pey) etc.
Thus
Therefore
81. Theorem. We have at once
r
j(; .a = {m[ — p) . (T = 2 . (ct / . e, (pe,) — (pe,) /. e.a)
r
(=1
2 ! Hi^ = 2 . /. c/, (p . j^'^a) = 2/ . e,, (pe^) / . e^ Ae^ Ae^ (pej
A:
= 2 . /. (pe^) e,^. /. e^. ^(pe,) ^e^ e^
= 2 . / . (pe,) ^(pej) 4e,. e^
= 2 ./. ei^e,. ^(pej) (pe,.)
Since ;^" . ct = (wig — p . 'x!)'^> ^^^ have
;jr". (y=-Z A .e-.ejAa (pe^ (pe^)
In general, we find
X^'\ a = S .Aeiej....e, Ja(pe,.) (pe,) .... (pej
82. Theorem. If we use the notation of the ^-pairs, these become
m[ = /^pO
»j
's = --,I.^,A^,....C.AipQ{pC^....ipQ
;K<»'.ct = -, J^,....^,^(T(p^,)....(pQ
In this form, the independence of the expressions m and x from any particular
unit-system is shown.
83. Theorem. Let us write further
m' (p, , p, . . . . p,) = -J 7^1 A^,.... ^,^(p, 4^,) (p„ Q .... (p_, Q
Then, from the properties of the ^'s, this form will reduce to
»''(pi, • ■ • • , pj = „, [»'i (pi) ^"i (p^) • • • ■ m^ (pj
2 . wi; (p,) (w{ (pg) .... m[ (p, 1 pj + . . . . ]
34 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
according to the rule : Insert m'l before every selection of p's taken according
to the partitions of s, giving each term of s — Jt, factors the sign (— )", and
writing in each factor the product of the p's in every order possible when the
p of lowest subscript is kept first in the product. For example, if s = 3, we
have the partitions 3 = 1 + 1 + 1 = 1 + 2—3. Hence,
7w'(pi, p.,, Pa) = Yl t"*! (Pi) ^'^ (P-') m'l (ps)
— m[ (pi) "»i (p2 p:)) — '"i' (pa) ^i (pi Ps) — "'i (p.3) '"1 (pi P2)
+ n'l' (pi P2 p3 + Pi p.3 pu)]
We note that
m' (pa p/, p, p/) = »"' (p/ pn p;, pc)
If s =: r + 1, this form must vanish identically.
84. Theorem. If we put
j(^{p,....p,)a = ^^^A.^,....^,Aa (p, ^i) (p. ^2) ■ • • • (ps Q
then, if Xi stands for [x (pi)] (o), X\'i ^o'" \x (pi P^)] <^» ^t<^->
;t(pi •••• p.)<^ = ^,|[%i-;t2 ••■• r^(<^) — 2.%i.%.3 •••• 3:s^i,.(<t) + ----]
The rule is the same as for the preceding expression of m\ thus
X (pi, p3, Pa) • <J = ^, [;ci . ;t3 • Iz (ff)
— ;i:i a:-'3 (<t) — Z3 ;i:i3 (<7) - xz xvz {<^) + {x\zz + ;i:i33) <y]
85. Theorem. If ???,,, ,,,...., ,, is the function 2 . y\' g-i ■ ■ • -gl' , summation
over all permutations of 1, 2 .... t, then
"».., ...... = /^'i^iV ■ • •?. ^ (p-^- ?i) (p^^ ^.)-- ■■{f'^t)
86. These numbers m and functions ;^ are called invariants of p, or of
Pj, p.j . . . ., as the case may be, since they do not depend on any particular
system of units. It is obvious that any function of pi, pg . • • ■ pt, containing
only ^-pairs, is an invariant^ in this sense.
87. Theorem. If p a = 0, then x' ■ u:- = »'i «, p ;|^' . a = w?, p a =
x" . a ^ Wo "-
^'** .a =■ m^ . a
In general, if p a = <; a, then
j^.a = {mi — g)a %'"' . a — (w, — m,^^ g -\- ±(7") a
If pay = g uj + ao p a^ = gf a,
;^w . a, = (m, — w«_i g + ....±g')a, — {m,_, — m,_., g.... Tj/""') a^
Similar results may be found for the other latent regions of p.
I ct. id'AuLAT 1.
THE CHAUACTERI8TIC EQUATIONS OP THE ALGEBRA 35
m. THE CHARACTERISTIC EQUATIONS OF THE ALGEBRA.
88. Theorem. Of Hh! miita taken to define the alj^ebra in the preceding
chiii)ter, certain ones will Ijo of pre-character a, po.st-cliaracter /?. Let the
number of such be represented by r/,,^ . Then the total nuinljcr of those of
post-cliaracter ji will be
It'll =7lifl + Ihn + + S^
The number of pre-character a will be
nl = «ai + «a3 + + n^v
89. Theorem. We may state the general multiplication theorem again in
the following form, ^ being any number :
V
where
k' — kt (iiy — /i > k' — k X ftij, — |U„j
In this equation each coefficient o is a linear homogeneous function of certain of
the coordinates a- of ^, namely those of type a-',;,"* where '^u,, combines with
°^^/, without vanishing.
90. Theorem. If we multiply ^ into each unit, and form the equations
resulting from the pre-latent equation^ of ^, say A' . ^ =■ 0, we have at once,
because the units have been chosen for the post-regions of a certain number ^,
The orders of these determinant factors are n'l, n'o .... n", their sum
being equal to r.
91. Theorem. An examination of the determinant A,' shows that it may
be divided into blocks by horizontal and vertical lines, which separate the
different units *^ii, ^^l-^, .... according to the power of 6^ which produces the
units, the order being
c , c
^ul • • • • skmjj
There are ^,, columns and rows of blocks. But, from the properties of the
coefficients a, the constituents in the first block on the diagonal are the only
constituents in any block on the diagonal. Hence we may write'^
a; = A;r» A^«3 .... A/^ic.
92. Theorem. The determinants A-,., s= 1 .... C(, are irreducible in the
coordinates of ^, so long as ^ is ant/ number. For, if one of these determinants
were reducible, then the original separation by idempotents could have been
pushed farther — as this separation was assumed to be ultimate no farther
reduction is possible.^
'On the general equaUon see Studt 3, 3; Sforza 1, 2; Scheffbbs 1, 2, 8 ; Molien 1; Cartan 2;
SUAW 4; Tabeu4; Fkobenius 14.
» Shaw 4. Cf. Caktan 3. « Ct. Caktan 3.
36 SYNOPSIS OP LINEAR ASSOCIATIVE ALGEBRA
93. Theorem. Confining the attention to A'^., lei the units °^ji whose pro-
ducts by ^ give A«., be 7i in number, with the pre-characters a= 1, ••••,/,
/^ h. The coordinates x appearing in the coefficients a, must be of the form
x'*'"'-'. It follows that if ^ be chosen so that all coordinates x not of these
characters (uia.,), aj, a^^ 1 ■•••/, are zero, then the value of A,';^ will not
be affected. The aggregate of such numbers, however, obviously constitute a
subalgebra which includes x„, a=: 1 ..../. These numbers, say ^''''^, when
multiplied together yield a pre-Iatent equation ^„„= 0, which must be a power
of A-^., and therefore irreducible. It follows that if we treat this subalgebra
as we have the general case, we shall find but one shear making up the whole
of each latent region. Consequently the units of this algebra take the form
e„,a, («!, a2= 1 /)
They may be so chosen that
The partial moduli are evidently^
e»,a, (ai=l .... /)
94. Theorem. Since any unit °£^ may be written e.^ °^^ it follows that no
expression e^,,. "^^ can vanish, else
Hence if there is one unit °^^, there are all the units'
95. Theorem. The units of the algebra may therefore be represented by
the symbols
e«3 e^y Ss
where the numbers e^' and e^j* are such that
f'a? '^yS <^^*: o'^/Sv t'ai
The numbers e^^ form an algebra by themselves, such that its equation consists
of linear factors only,^ as
A, = {a^ -
96. Definition. An algebra whose equation contains only linear factors
will be called a Scheffers algebra. If, further, it contains but one linear
factor, it will be called a Peirce algebra. If it contains fiictors of orders
higher than unity, it will be called a Cartan algebra. An algebra consisting
of units of the type e[^l only, will be called a Dedekind algebra.* The degree
of an algebra is the order of its characteristic equation in ^.
' Moi.iEN 1 (urspriingliche systeme); Cartan 1, 3; Suaw 4; Fuohenius 14.
' Cautan 2; Fkobenius 14.
^Caktan 1, 3. On tbe "multiplication" of algebras by eacb iitlier, see Ci.iffoud 8 ; TAiimt 1;
ScHEKPEii.t ."!. Cf. Taiiku 4; Hawkes 1, 2; Fkoiienius 14.
•On classification see ScuEFKEUS K, 4; Moi.ikn 1, 2, 3 ; Caiitan1,2; Siiaw4; B. 1'k)U(:e, 1, .S,
THE CHARACTERISTIC EQUATIONS OF THE ALGEBRA 37
97. Theorem. Let the algebra be of the Schefi'er's type. The irreducible
factors of it.s pre-latent equation are all linear; hence in the latent post-region
of any root of ^, the shears are of wi'llh unity only. The units defining ihe
i-ih region become
"^jt a= 1 . . ■ . p j= I .... Ci 1= I .... /j^j
Ihi > t^is >l"'Oi
The product of ^ into any unit is'
S • S.H — — (*fjk--k KflC
where
Ic' — hlO fi^j, — /.- > k' — /.- ;: fii,y — (I, J f -1 j if Id = h
98. Theorem. If we remove from this algebra all idempotent units, the
remaining units form a nilpotent algebra of r — ^j dimensions. The equation
A' ^ = reduces in this case to a determinant whose constituents on the
diagonal and to the right of the diagonal all vanish, hence it is evident that
the product of any two of its numbers is expressible in terms of at most
r — p — 1 numbers. Let the original units be <^,,+i, <?»p+2 ■ • • • 4'r- Then the
products ^,,4),.. do not contain a certain region defined by a set of units
%^-i ■■■■ ^p+h, {h >o)
The products of these h^ units (which constitute the region fj, let us say,)
among tliemselves and with any other units, are linearly expressible in terms of
%+in+t (< = 1 • • • • r — p—h,)
Similarly any product <^,^ ^,., <^,^ can not contain a region e^, defined by
Hence {e^.Jfal-, |%! ■ jfij-, ^"^ {f^} -Ifsf depend only on <^p^t, < ^/fi^ A,.
Proceeding thus, it is evident the domain of the nilpotent algebra may be
separated into regions defined by classes of units which give products of the
form
\^c\ . \ej\ = \e,] {k>i, k:>j)
In particular, the units of the Scheffer's nilpotent algebra may always be
chosen so that, if they are >:,, yj^ . . . ., then
>7i Tj = 2 Yijk >7t {k > J, k >y)
It is also evident that for any r — ^) + 1 numbers ^t we have
The* products of order I form a sub-algebra of order r',
r' < ;• — Z -h 2
'Shaw 4, 5. -'^ScHErFERS 3 ; Cartan 3 ; Shaw 5 ; Frobesius H.
3
38 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
99. Theorem. In any Cartan algebra the units may be so taken as to be
represented by
e% {i=l .... p a,(3=l.... tCi)
yii'ii (*, y = 1 — i> «; i^ = 1 — w'i)
The laws of multiplication^ are
100. Theorem. Returning to the Scheffers algebra, if we retain only its
nilpotent sub-algebra and the modulus, we shall have a Peirce algebra. The
equation of this algebra will contain but a single factor and the pre- and post-
characters of its units may be assumed to be the same. The nilpotent 6 becomes
the sura of the nilpotents Bi + do. . . . + 6^,. The product of ^ into any unit
may be written^
S • hjk = 2 dfjki-k hj'k'
k' — klO ny — k>k'—ktiiy—Hj j'lj\ih' = k
101. Theorem. Let the characteristic equation of any number be
^- _ /j . ^'"-1 + ....+ (-)»'/™ = {m<r)
where f^ is a homogeneous function of the coordinates of order i. Differen-
tiating this equation, and remembering that d^ is any number, we arrive at m
general equations connecting 1, 2, . . . . , m numbers of the algebra : as
{^f-'^. + ^'r'^2^. + • • • •?2^i"-^)- [/i [Q^T'
(^r-^^2^3 + ----)-etc. = o
These equations are the second, third, etc. derived equations of the algebra,
according as they contain two, three, etc., independent numbers ^i, ^2>
etc. These equations lead to many others when the scalars of ^ are intro-
duced.^ The new coefficients fi{<^a,....'(a.^ will be called the scalar charac-
teristic coefficients of order i for ^„, .... ^„.. They usually differ from the
coefficients m.
102. Theorem. The general equation of r numbers of the algebra of order r
is written (2 representing the sum of the r! terms got by permuting all the
subscripts)
'CAKTAK8. »8haw4, 6. sTabbk, 2, 3. Shaw 4.
TUK CHAKACTKUI8TIC EQUATIONS OF THE ALOEBUA
39
In this equation, omitting the Kubscript 1, so that m = mi
^i (^t, Q =m'(, . m'(j — m ^, ^j = m, {'(j, Q
"t3(^(, (j, </.) = "J^i • '"^J • '»^* — "»Ci • ^^j '(k — mi^i . m^i^^fc
= w'a (?(, ?*, ^j) = "'3 ( ^^> ^*i Ci)
These formuIaD Ibllow from the identities
«'"«(<?> I, <?>1 •• •^l)=»'l ['"»-! (4*1 ;<?'l ••■<?'l) ■1>l — ^^-Al>l ■■■1>l)-1>l
+ ... + (- !)"-> »n. («?., <?>,) . <i>r- + (-- 1)" '"i <?>. . rr' + (— 1)"+' . <?>;]
and
w, (<?)i ...<?>,:...<?»,...<?).) = m. (<?)i ...<?),•..<?)(••■ <?>») i, y = 1 . . . »
We arrive at the formulae directly by differentiating
s! m,{^i, <^, <^,) =
m(p^
1
m<pi
m^f
2
mfl
m^i
m(
m^r'
m^r'
m
w^;
mrr'
m
h»-3
3
TO<^,
103. Theorem. A study of the structure of all algebras of the Scheffers
type gives u.s the structure of all algebras of the Cartan type, as we may pro-
duce any Cartan algebra by substituting for each partial modulus of the
Scheffers type a quadrate, and then substitute for each unit of the algebra a
sub-algebra consisting of the product of this unit by the two quadrates which
correspond to its characters.*
104. Theorem. Each Scheffers algebra may be deduced from a Peirce
algebra by breaking the modulus up into partial moduli, accompanied by
corresponding separations of the units. For, if all partial moduli of a Scheffers
algebra are deleted from the algebra, leaving only the modulus, and a set of
nilpotent units, we have a Peirce algebra. Any Peirce algebra may be con-
sidered to have been produced in this manner, so that to any Scheffers algebra
corresponds a Peirce algebra, and to any Peirce algebra correspond a number
of Scheffers algebras.
105. Theorem.
If the characteristic function of an algebra be
AJ>
..a;;i.= o
wherein A, is a determinant in which ^, the general number of the algebra,
occurs only on the diagonal, and the other constituents are linear homogeneous
functions of the coordinates oft,', and if we substitute for ^ where it occurs 4-1
'Cartan 3. Cf. Molien 1; Shaw 4.
40
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
any arbitrary number of the algebra, then the resulting expression may be
written 0(4') = Alj'' (4-) AS= {4') ■ ■ ■ • ^pp{'^)- This expression will vanish only for
wherein K^(^i has the meaning given in part II, chapter XIX, art. 3.
Thus the algebra whose characteristic equation is
(^0^0
•'oo
^10
0-0 =
gives the expression
^u-4
This expression vanishes when and only when -^=-21, Kq\, or q^; wherein
qz ^^ ^a ^^33
That is, the expression is factorable into (1^ — q^) {'^ — /ij,) (1^ — q.?j.
As a corollary, the expression
«oo —
«10
Q
OqI • • • -^On-l
«n — 6 «i n-i
«n-10
a„_ii a„_i „_i — 6
IS
factorable in the matric range ofq^, Kq^ .... /l" ' (71.
rV. ASSOCIATIVE UNITS.
106. Definition. The multiplication formula in § 100 may be used to intro-
duce certain useful new conceptions. It reads
Jd — hio
Ki
h>k' ~ k^ixj,~fij
j' < J if k' = k
Let us consider an algebra made up of units which will be called associative
units, represented by /l„(, such that
where
'k' Citjj, . t j /;! _|. ^.
k^O i >y if /.- =
c= 1 if (Uj > A; ^ jUj — fij,
c = if ju, < /c < ^/j — fij,
Since there is a modulus Cq, and since £«-, = ^/k'Co, every unit ^j,^, is expressible
as a sum of these units 2,^,; multij^Hed by proper coefficients, and every number
'( is expressible as a sum of the units with proper coefficients. Hence, we may
express ^ in the form
l^i > ^'^ " /<,
k^ i ]>y when /•
ASSOCIATIVE UNITS 41
The Peirce algebra i.s expressible therefore as a sub-algebra of the
algebra of the af^sociative units whose laws of multiplication ' are
where
Uj >■ /c ^ jU( — jUj Jc : i >-y when 7c:=
fXi > // ^ ^i — fi), k'^ i >/ when lc'=
c= 1 if ^i>h+ k> l^i— Hj, k + /^' = /■ >/ when k +/.-'=
c = if /i( :;, /.; + k'<^^i — ^j,
107. Definition. An expression of an algebra in terms of associative units
will be called a canonical expression. In many cases the associative units are
the units of the algebra, in part at least, but the units of the algebra will
frequently occur as irreducible sums of these units with certain parametric
coefficients. This theorem extends C. S. Peirce's theorem that every linear
associative algebra is a sub-algebra of a quadrate" of order r^.
108. Theorem. The Schelfers algebras derived from this Peirce algebra
have partial moduli of the form
«t = ^hho Oi='^ ■■■■ Oi
When each partial modulus ei is of the form ?.jio, the SchefTers algebra coincides
with the algebra of which the Peirce algebra is a sub-algebra. Such SchefTers
algebras will be called primary algebras. The units in any Scheffers algebra
are separable into classes according to their characters, those of chai'acter j
having in their expression units X of the type
7.,j.„ or %j.,k j\=l (Ji
109. Definition. The units of a SchefTers algebra are separable into those
of characters, (aa), and those of characters (a/3), a :^ ^S. Those of characters
(aa) constitute the direct units. Those of characters (a/3) are the skew units.^
110. Theorem. The pre-latent (post-latent) equation must contain the
factor (aj.o — ^) to that power which is the sum of the multiplicities belonging
to i:
The characteristic equation will contain {a^Q — ^) to that power which equals
the maximum multiplicity^ fif\
111. Theorem. A Cartan algebra will have for a canonical expression
^ ^ 2 a^k ?'-a/30 ^ijk
where the units Z and ?J are independent of each other.
I Shaw 4, 5. sc.S. Peikce 1, 4. 'Schkffers 3. < Scheffers 3; Shaw 4.
42 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
112. Theorem. We may obviously combine these forms into still more
compound expressions as
^ = 2 a(ij,ki)iuhh) • • ■ • ■^'id,k, '^"uj,k, ■ ■ ■ ■
Such numbers are evidently associative, and could be considered to be the
symbolic product of algebras with only one 2..
113. Theorem. Returning to the equations of the algebra §108, we see
they evidently depend on those associative units which are of weight zero.
The equations are
characteristic : = A . ^ = Aj" AJ^ .... A^pi
p
pre-latent : = A'. ^ = 11 . A; ^'='
i = l
P
post-latent:^ = A". ^= 11 . Aj^=i ^'
1=1
114. Theorem. The number Aj (^) can not contain any associative unit of
the form /ljj,o> where the constituents of Aj are of the form- ajj,o>
/j = 1 ....</!. The factor A, (^) is the i-th shear factor off.
115. Theorem. The product Aj f . A3 if can xiot contain any associative
unit of the form X^^j, p; or X;„^,,,. The theorem may be extended to the
product of any number of shear factors.-
116. Theorem. The product (Aj f)'" can not contain any associative unit
of the forms
'^j'liiOj \y, 1 • • • • X;,j-, m_i
117. Theorem. The third subscript in X^^^, h, is called the weight of X.
Every number f may be written in the form
f = f (») + f (») + •••• + f <"" a i 0, i > a
The weight of f is the weight a of its lowest term. The weight of the product
of two numbers is the sum of their weights.
118. Theorem. The terms f** constitute an algebra. This may be called
a compa7iion algebra, and may or may not be a sub-algebra of the given
algebra.^ The quadrate units of an algebra evidently belong also to the
companion algebra.
119. Theorem. To every transformation of the units of a companion alge-
bra corresponds a transformation of the units of the given algebra. Hence
'Cartan2. 'Shaw 4.
^ Cf. MuLiEN 1. "Begleitende" systeniB include these companion algebras, and may or may not be sub-
algebras of the given algebra.
ASSOCIATIVE UNITS
43
the ^*'" terms may jilwayH be taken according to the simplest form for the
companion algebra,'
120. Theorem. If the general equation of an algebra is
'C — Wi ;"■ ' + Wg ^'•-2 =0
a
and if when ^ = 2 a-jC?, we put y = S . e,-^ — , then v • ^a = gives r erjua-
tion.s, not necessarily independent, from which the r coordinates may be
expressed linearly in terms of rj arbitrary numbers. These determine the
nilpotent system; or from the r — Ti coordinates which vanish, the Dedekind
sub-algebra."
121. Theorem. Since v = ^ -^^ V, and /^ v • p = ^> therefore
V . «', (p) = V • H. (p^i) = ?. . /^2 V • /^i (pCi) = ^2 /^i (^^i)
But /. i,'i(^a^i) = nii{(^.,), therefore we have
V J"! (p) = ^2 "ii {Q
mi (e<) = t = 1 . . . . r
V m, (p) = ^3 /^i A^, A [(^3 ^,) (pC^) - (^3 ?2) (p^'i)]
= 2^3
This can vanish only if
Again,
hence
= 2 :£ e
/^2(^3^l) /^3(P?2)
i = l
r
= 226; [wii (Ci) . ?Hi (p) — mi . (e; p)]
This vanishes if, and only if,
or
2 a;^ ] mi (e,) m, (cj) — m^ (e, Cj) } =
t = 1
t= 1
These are the equations referred to in §120. The method used here has an
obvious extension.
' Of. Shaw 5.
'Cartas 2.
44 SYNOPSIS OP LINEAR ASSOCIATIVE ALGEBRA
V. SUB-ALGEBRAS. REDUCIBILITY. DELETION.
122. Definition. A sub-algebra consists of the totality of numbers ^ such
that
^ = Xxf Ci i = 1 . . . . ?•', r'<^ r
for which^
?i ^2 = 2 . x[x'! y^k Ck i, j,1c = \ r'
123. Theorem. In a SchefFers algebra all units with like pre- and post-
character (aa) define a Peirce sub-algebra.^
124. Theorem. The Peirce sub-algebras formed according to § 1 23 define
together the direct sub-algebra. The characteristic equation of this sub-algebra
does not differ from the equation of the algebra.^
125. Theorem. The quadrates form a sub-algebra, the semi-simple system
of Cartan/ called a Dedekind algebra.^
126. Theorem. All units in a Cartan algebra with characters chosen from
a single quadrate form a sub-algebra, the product of the quadrate by a Peirce
algebra. Its equation has but one shear factor.
127. Theorem. All sub-algebras of §126 determined by the different
quadrates form the direct quadrate sub-algebra. Its equation does not differ
from that of the algebra.
128. Theorem. All numbers which do not contain quadrate units form a
sub-algebra called the nil-algebra (Cartan's pseudo-nul invariant system).*
The units of this system are determinable to a certain extent (viz. those which
also belong to the direct sub-algebra of § 1 27) from the equation of the
algebra. The other units are not determinable from the characteristic
equation of the algebra.^
129. Definitions. All numbers ^, which are expressible in the form
r'
i = l
form a complex. The entire complex may be denoted by E^, E.,, etc., E=E(,
denoting the original algebra."
The product of two complexes consists of the complex defined by the
products of all the units defining E^ into the units defining E^, indicated'' by
E, . E.-,
An algebra E is reducible when its numbers may all be written in the
' On the general subject see Study 1, 2, 3 ; Scheffebs 1, 2, 3, 4, 7 ; B. Pbiboe 1, 3 ; IIawkes 1. 2.
'ScHEFFEKsS. Caktan 2. 3 Shaw 4. ■» Cahtan 2. i' Fkodknius 14.
• Epstken and Weddehburn 2. ' Epsteen and Weddbrburn 2 ; Fkobenius 11.
SUB-ALGEBRAS. REDUCIBILITY. DELETION 45
form C = ^'i + C: where ^| belongs to a comiilox E^, ^.; to a complex E.,, such
that,'
E,.E^= A', A', . E., = E^. Ei=zO E.,.E., = E.,
An algebra is irreducible when it can not be broken up in this way.
When reduciljle into a complexes we may write
E=E, + E,+ .... +E^
130. Theorem. An algebra is reducible into irreducible sub-algebras in
only one way.^
131. Theorem. The necessary and sufficient condition of reducibility is
the presence of A numbers e^ .... e,,, such that if ^ is any number,'^
^e^ = eX e^ = e. e. e^ = e^ e, = a = 1 h, a^^
132. Theorem. The characteristic function of a reducible algebra is the
product of the characteristic functions of its irreducible sub-algebras." The
order is the sum of the orders of the sub-algebras, and the degree is the sum
of the degrees of the sub-algebras.
133. Definitions. The region common to two regions, or the complex
common to two complexes Ei, E.,, is designated by E^.,. If the complex E^ is
included in the complex E.^ this will be indicated by* E^ 1 E.,.
The reducibility used by B. Peirce is defined thus, E is reducible^, if
E=^E, + E. Ell El EI<Eo EyE.,<Ey, E.E^lE,.
An algebra is deleted by a complex E.^ if the units in E.^ are erased from
all expressions of the algebra, including products. The result is a delete
algebra, if it is associative. It may not contain a modulus however.^
134. Theorem. Let the product of ^a be given by the equation
l....r r
^0 = 2 a-i yj y^k e^ — 2 x'^e^
ij,k k = l
If the units may be so transformed that the product may be expressed by
means of the equations
l....r'
x'i = 2 Xj 7/^. y.j^. 1=1 r' r'<^r
i,k
\....r
x\,= 2 Xjy„Yi-J>' i'= r' + 1 /•
then the units e^ . . . . c^,, define a delete algebra,® called hereafter a Molien
algebra. If an algebra has no Molien algebra, it is quadrate.
'See refereuees § 122. 'Scheffers 3, 4. 'Epsteen and Wedderbcrn 2.
*EpsTEEN and WEDDERBrRN 2. On the definitions of reducibility see Epsteex and Leonard 3;
Leonard 2.
'Scheffers 3, 4; Hawkes 1, 3. Cf. Moi-ifn 1 ; Shaw 5.
•Molien 1. This is Molieu's " begleitendes " system.
46 SYNOPSIS OP LINEAR ASSOCIATIVE ALGEBRA
135. Theorem. A Molien algebra of a Molien algebra is a Molten algebra
of the original algebra. Two Molten algebras which are such that the co-
ordinates of the numbers of the two algebras have q linear relations, i. e., whose
numbers are subject to q linear relations, possess a common Molien algebra
of order q, and conversely. If the Molien algebras of an algebra have no
common Molten algebras, then the numbers in the different Molien algebras
are linearly independent.'
136. Theorem. If the complex of the linearly independent numbers of the
form ^o — a^ be deleted from an algebra, the remaining numbers form a
commutative algebra.^
137. Theorem. If the commutative algebra of § 136 contains but one unit
the original algebra is a quadrate.^
138. Theorem. If the delete algebra in § 136 contain more than one unit
it may be further deleted until the delete contains but one unit. This unit
will belong to a quadrate algebra which is a delete of the original algebra.^
139. Theorem. The scalar of any number contains only coordinates which
belong to the units in the commutative delete alg' bra.^
140. Theorem. The pre- and post-latent functions of a delete algebra are
factors of the corresponding equations of the original. The characteristic
equation of the delete is a factor of the characteristic equation of the original.^
141. Theorem. The two equations of a quadrate delete algebra are powers
of the same irreducible expression.'
142. Theorem. An algebra is a quadrate if its characteristic equation is
irreducible and if the scalar of any number contains only coordinates belonging
to the units of the quadrate (which may be a delete algebra).'
143. Theorem. The irreducible factors of the characteristic equation of an
algebra are the characteristic functions of its delete quadrate algebras.'
144. Theorem. The number of units of a delete quadrate is the square of
the order m, of its characteristic equation. If they are e^j, then
Cij e^i =■ ^jk ea i, j,h,l=l m
The delete quadrate is also a sub-algebra of the original."
145. Theorem. If, in a SchefFers algebra, the product of ^ into and by the
units er,e,._, er_„, vanishes, provided ^ is not a modulus or a partial
modulus, then the algebra may be deleted by the complex of e,. Cr - r,- The
' Molien 1.
'Molien 1. Molien points out that tlie unite may be claasifled according to their quadrate character,
thus approaching Cartan's theorem, J 99.
SUB-AIXJEBRAS. REDUCIBILITY. DELETION 47
delete algebra will have an equation with all the factors of the original algebra,
but each appearing with an exponent lees by unity for each deleted direct unit
belonging to the factor.'
146. Definition. The deficiency of a Peirce algebra is the difference between
its order and its degree."
147. Theorem. The units of a Peirce algebra may be so chosen that, if it
is of deficiency h, one unit may be deleted, giving a delete algebra of deficiency
h— \, which is a sub-algebra of the original.^
148. Definitions. An algebra E is semi-reducibJe of the first kind when it
consists of two complexes, E^, E.^^ such that,''
E^Ey<E, EiE,<E, EoE^<Ez E^E^iE,
Am algebra is semireducible of the second kind when it satisfies the
equations*
EyE^lE^ E\E,<E^ E^E^ = Q E^E^^E.^
If in any algebra
EiE^lE EiE.,<E,^ EoEi<Es E^E-^lE.;,
then Eo is called an invariant suh-algebra}
149. Theorem. \^ E has an invariant sub-algebra E.,, the algebra K pro-
duced by deleting E., is a delete of E, called con^plementar y to^ E^.
150. Theorem. If ^, is a maximal invariant sub-algebra oi E, and if there
exists a second invariant sub-algebra E., o{ E, then either E is reducible or E.^
is a sub-algebra ^ of Ey.
151. Theorem. If £", and E., are maximal invariant sub-algebras of £", and
if E^o -^ 0, then Ey^ is a maximal invariant sub-algebra of both Ey and E.^.^
152. Theorem. A normal series of sub-algebras of E, is a series Ei, Eo, ....
such that jB", is a maximal invariant sub-algebra of £"8 _ J (-ffo^-E"). lfKi,Ko,. . . .
are the corresponding complementary deletes, then apart from the order the
series /fj, K,, .... is independent of the choice of J^i, E^, . . . .*
153. Theorem. Let «, be the order of ^,; 7,, the difference between aj_i
and the maximal order of a sub-algebra of £"4 _j which contains iS", ; /.•, =a,_j — a,.
Then the numbers /j, L, .... are independent of the choice of the normal series
apart from their order. A like theorem holds for l-i, K, . . . .*
154. Definition. An algebra which has no invariant sub-algebra is simple}
>S0BEFFER3 3. > StaEKWEATHBR 1. > EPSTBKS 1, 3.
* Epsteen 1. s Epsteen and Weddbrbcrn 2.
48 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
155. Theorem. The complementary deletes K^, K^, . . ■ ■ are all simple.'
156. Definition. The series E, Pj, Po, .... is a chief or principal series
when Pj is a maximal sub-algebra of Pg_i which is invariant' in E.
157. Theorem. The system of indices of composition is independent of the
choice of the chief series, apart from the sequence.^
158. Theorem. An algebra is irreducible if its quadrates may be so arranged
Qi,Qz- • • • Qp that there are skew units of characters (21), (32) ... . (^^jp — 1).^
VI. DEDEKIND AND FROBENIUS ALGEBRAS.
159. Definition. A Dedekind algebra is one which is the sum of quadrates
Qi, Qz Qi,- Its order 3 is r = ?q + + "7, •
160. Theorem. A Dedekind algebra has a sub-algebra of order A, whose
numbers are commutative with all numbers of the Dedekind algebra. No
other numbers than those of this sub-algebra are so commutative.*
161. Theorem, A Dedekind algebra is reducible and the sub-algebras are
found by multiplying by the numbers e„, a = 1 . • ■ • /;, in terms of which the
commutative sub-algebra may be defined, [e^e^ = S^^^e^].^
162. Theorem. The characteristic equation of a Dedekind algebra is
Ai A. . . . A„ = 0. The pre- and post-equations ^ are AJ'' A^"= AJf" = 0.
163. Theorem. If a Dedekind algebra has only linear factors in its equation
it is a commutative algebra.^
164. Theorem. The scalar of e„ is given by the equation
The scalar within a single quadrate, Qi, may be indicated by S.^. For any
number we have ^
i
165. Theorem. An algebra is a Dedekind algebra when in the general
equation, vu, the coefficient of ^'' ^, contains each coordinate in such a way
that the equations
1=1
dm^^
give " a:i =....= ar, =
> Epsteen and Weddekdukn 3. sScueffers 8, 4.
»Cf. Frobesius 14. Cabtan 2. This la Cartau's semi-simple algebra.
< FKonENirs 14. He cails tliese invariant numbers. '' Frobenius 14.
•Cautan 3, see §121. Evidently | m, (e., «^) | ±0.
DEDEKIND AND FROBENIUS ALGEBRAS 49
166. Theorem. If Aj is the deluniiiuant shear factor corresponding to the
quadrate Q,, then Si . A, = for all numbers of the algebra, and if e, is the
partial modulus of this ([uadrate,*
ei Ai = A. <-! =
The i + 1 scalar coefficient of any numbers vanishes ; i. e.
167. Theorem. If A, («) = Aj [b) then for a determinate number^ c
c~^ac = /j
168. Definition. A Frobenius algebra i.s one which can be defined by r
numbers o^ . ■ . ■ o, which .satisfy the equations
oT' = ^0 = Oi i=l r
Oi Oj = Ok ur^o^ = Oj Oi = o^oj- • hj=^ r
o, Oj . 0,, = 0i .o, o„ i, j,k=\ r
The multiplication table of these units defines a group, and any group of finite
order or infinite order may be made isomorpl)ic to a P'robenios algebra.''
169. Definition. Two units o,, Oj are amjugate if for some determinable
unit 0^. ,
Oi = o„Ojor^
If we operate on Oj by all units of the algebra, Oj . . . . o^, we arrive at r
different units as results. These are said to constitute the ^th conjugate class.
There will be k of these classes. Also /^ is a divisor of r.
170. Theorem. For each unit in a conjugate class we have (as Oj is the
modulus or not) :
S . 0^ Oj oi~^ = S . OjZ= 1 or
171. Theorem. If the sum of all the units in the ^th conjugate class be Ix',,
then for any unit
A'( Oi = o,Kt i=l r
There are k different numbers Kf, K^ . . . . K,,.
172. Theorem. The Ic numbers K^, < = 1 . ... 1c constitute a commutative
algebra of A; dimensions, that is
173. Theorem. We have (according as o< is not or is the modulus) :
S . Kt = -^ S . lOi Of of ^ = rtS .o, = Q or r^
174. Theorem. A Frobenius algebra is a Dedekind algebra of k quadrates.
The k numbers Ki determine the k partial moduli, one for each quadrate.
' FaoBENins 14. Shaw 4. = Frobenius i4. Other theorems appear in Chapter XIX, Part II.
sFrobenicsS, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18; Dickson I, 2, 3, 4; Burnside 1, 5; Poincabe 4 ; Shaw 6.
50
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
The widths of the quadrates being represented by w,, i = 1 .... k, we have
Kt^'^gu^i i,t=l k
175. Theorem. It follows that if we take scalars
r . SKi = r = wl + ii4 +wl
r . SKj = = wlgji + tdgj3 + m9jk i = 2 k
176. Theorem. Let the scalar of o< in the quadrate i be represented by
AS'foj, orsf, then
k
2
i = l
^.0,= 2 M;f./S<>,= 2 toW?
k
2
177. Theorem. We have
^« K, = g,, ^« e, = g,, = r, . S'^ o, = r, *'«
Hence ^ _ ^^ ^(^ + m,| ««>+.... + li'X «f '
= «-js;.'> + M;|«f +....+ <4«r j=2....k
If we write for icj sj"' the symbol xf} (called by Frobenius the i-th
characteristic of Oj) we have
_ -. _A _ fi)
"'J — ^ ^ — Xi
where A is the determinant \x\\ xf ■ ■ ■ ■ X<*'l ^"*^ ^j '^ ^^^ minor (including
sign) of tc^. This determinant A evidently cannot vanish.
Also
Ki = y-t 2 e^
<=i
Xt
rt'Zeis'i
(t)
anc
(2^,f=r2ir,
178. Theorem. Hence
s? K,
/rj . . . . «[*■'
. . . si*'
4" K„lr^....sf^
-=-
41' S«> .
• • • sf
4" ^./n- • • • ■ «lf^
4'> 4^' •
...sr
MJ,
;ti'^ ;tf • • ■
or
or
179. Theorem. For all values of a, h
d = l ^h
rs'^ 6'«' = 2 S^'^ 0,0^0,0;^'
d = l
— -y*"' y«> = to, 2' y">
~ Xa Xl> — "'i -^ Xac
' b c
where 2' takes o^ over the rj, values in the conjugate class* of 0^.
c
Also' w,wj 8^^ s^J^= 2 rf,^C' or x"^ x'i' = 2 ci'i>;ta'
u=l
« = 1
' See references to § 168. These apply to theorems following.
'BCRNSIDE 5.
DBDEKIND ANIJ FROBENIUS ALGBBKAS 51
180. Theorem. 2 . A"" . o^o^^ S"' o^ = -^ ^'^ o,
b=i ^i
181. Theorem. k . A"" . o^o^^ A"^' o,, = t :^:y
b=l
182. Theorem. S /S'<« o^ . S'^^ 0,7^ = -4-
183. Theorem. i *9"> o„ o;;' *S"" o, o,, = - ,- /S"« . o„ o,
h=i "'i
184. Theorem. 2 /S'">. o"' oz^ o„ o^ = -3-
a,fi=l ^i
185. Theorem. If o, i.s an independent generator of the group of units,
Off ... . Or-i) '•'^^^ ^^ we form the t-th LaGrangian of Oj, that is,
where u is a primitive mj-th root of unity, and »ij is the order of Oj {o^' = o^)
then for any number of the algebra, ^, we have a product
such that all numbers of the algebra are separable into wJ] mutually exclusive
classes of the forms (where it is sufficient for ^ to be any one of the units Oj
when the group is written in the form OjO\).
Uu {t = \ ....m,)
For ^/it, we have*
186. Theorem. If 0., is a .second independent generator, then we may
determine the equations oi o.;^ f^i {t =■ 1 . . . . m^. The latents Zi, determined
as in §48, used as right multipliers, separate the numbers of the algebra into
mutually exclusive classes, such that if these latents are /^^^^ , tlien (if
u :{: u', t 4: t')
Sfuufuu -— ^/uu S/Ku fu'u' ^—
This process of determining latents by the independent generators may be
continued until they are in turn exhausted.
187. Theorem. The ultimate latents are scalar multiples of independent
iderapotents of the forms /IJ'J,, where i= 1 . . . .u'l; s ^= 1 . . . .A: Multiplication
right and left by these idempotents will determine every quadrate unit P.,^^,
i, y := 1 . ■ . ■ u\; s = 1 .... A", in terms of the c generators Oj . . . . o^.
188. These results may be extended easily to cases in which the
coefficients of the units Oj are restricted to certain fields.
I Shaw 6. This reference applies to §S 186, 187.
52 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
Vn. SCHEFFERS AND PEIRCE ALGEBRAS.
189. Theorem. Every SchefFers algebra with h partial moduli has h sub-
algebras, each with like pre- and post- characters.
190. Theorem. The general equation of a Scheffers algebra of h partial
moduli is of the form'
n . (aj — <f)"ii + "i2 + •••• + «ft =
191. Theorem. Every number of a Scheffers algebra satisfies the general
equation of its direct sub-algebra, which is
n (a,-0"« = o
This equation is the intermediate equation of the algebra.
192. Theorem. The characteristic equation of a Scheffers algebra is
n (a,-0''' =
1=1
193. Theorem. A Peirce algebra may have its units taken in the form^
cj<^ s = \ ....p t = 0.... ^,-1
194. Theorem. Units containing 6', t > 0, may be deleted, and the
remaining numbers will then form a companion delete algebra, called the base
of the Peirce algebra.^
195. Theorem. Any Peirce algebra may be made to serve as a base by
expressing its units in terms of associative units of weight zero.^
196. Theorem. The product of two units follows the law ^
C ff' C fit'' — y/i C fit'" /"' > fl A- i'l
197. Theorem. A Peirce algebra of order r, degree r, is composed of the
units ''
These have been called by Scheffers, Study algebras.
198. Theorem. A Peirce algebra of order r, degree r — 1, is composed of
the units
Cj = Xlio + >^220 ^S — - '^aiO "H "^12r-2 ^3 ^ '^111 H" "'^IS r— 2 ^4 — " ^^112 • • • • ^r -— '^ll r-2
'Cartan2. »Siiaw5. Of. Strono I. ^SiiawS. •'Shaw 5. Cf. Sciikffers 3 ; Caktan 3.
' B. Peikce 3 ; ScheffersS; Hawkks 1 ; Shaw 5; Stddy 3.
SCHEFFERS AND PEIRCB ALGEBRAS
53
This is reducible, if a and b do not vanish, to the case of a = 1 = /--.
If a = 0, we may take b—\ or 0. If h — 0, we may take a = 1 or 0.
When r = 4, eitlier a = 1, /> has any value ; or a = 0, /^ = 1 ; or a = 0,
?> = 0.
If r = 3, a = 0, h = 0.'
199. Theorem. A Peirce algebra of order r, degree r — 2, is of one of the
following
case
The ?. wi
When
(1). ^0 =
(11). e,=
(12). ej =
(13). e,=
(14). e,=
(2). e„ =
(21). e,=
(22). ei =
(23). ej =
63 =
(24). e,=
63 =
(25). e, =
(26). e, =
(27). e,=
(?8). e,=
(29). e
ypes." Only the forms of e^, e,, e., eg, e^ are given since in every
^5 = '^ns • • • • ^r-2 == ''•11 r-4 ^r - 1 = ''-H r - 3
be omitted iu each case.
r> 6.
110) + (220) + (330), type of algebra (*, ^^y,/ /-^)
210) + (320) + (13 r — 2) e, = (310) + (12 r — 2) e3 = (lll)
e, = (112)
210) + (320) ('3 = (310) e3=(lll) ^^ = (112)
210) + (320) + (13 r — 2) c„ = (310)+ (12 r— 2)
111)+ 2(13 r — 2) e, = (112)
210) + (320) e,= (310) 63 = (111) + 2 (13 r- 2) ei = (112)
110) + (220), type of algebra {i,j, ij,f j''^)
210) + (12 r— 3) e. = (211) + (12 r— 2) eg = (ill) + (221)
e,= (112)
210) + (12 r— 3) 6, = (211) + (12 r — 2)
111) + (221)+ 2(12r— 2) ^^ = (112)
210) + (12 r — 3) e2 = (211) + (12 r— 2)
lll) + (22l) + 2(12r— 3)+2c(12r— 2)
64 = (112) + 4 (12 ?•— 2) c = 0ifr4:8
210) + (12 r— 3) + (12r— 2) Co = (211) — (12 r — 2)
HI) — (221) — 2(12>-— 3) e4 = (112)
210) + (12 '- — 3) e, = (211) — (12r— 2) 63 = (HI) — (221)
. e, = (112)
210) +7i(12r— 2) 63= (211) h=0 or l'\i r:^l
lll) + 2ic(221) + 2 (2-c) (12r— 3) e, =(112) + 4 (11 r— 2)
210) + (12 r— 3) 6, = (211) — (12 r— 3) 63 = (111) — (221)
e, = (112)
210) + A (12;-— 2) e,= (211) 63 = (ill) + (221) + 2 (12 r — 3)
64 = (112) A=0orlifr:^7
210) + (12 r— 2) Co = (211) 63 = (111) + rf (221) e, = (112)
'B. Peirce 3; Scheffers 3 ; Shaw 5.
4
2 Starkweather 1. Cf. Shaw 5.
54
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
(2a). ei =
(2y). e,=
(2f). e, =
(27,). e, =
(3). Co =
(31). e,=
(32). ei =
(33).
(34).
e, =
ci —
e, =
(35)
(36). e,=
(37). .,=
(38). e, =
(39). e, =
210) 6,= (211) e3 = (lll) + c? (221) C4 = (112)
210) + (221) + (12 ;•— 2) 6. = (211) e3=(lll) 64 = (112)
210) + (221) 62= (211) e3 = (lll) e<=(112)
210) e, = (211) e3=(lll) + 2(12r— 2) 64= (112)
2l0) + (12r— 2) e2 = (211) fg = (lll) + 2 (12?— 2) e4=(112)
210) + (12 ?•— 2) e2 = (21l) e3 = (lll) ei=(112)
210) 6,= (211) e3 = (lll) e4 = (112)
110) + (220) + (330), type of algebra, {i,j, k, 1c . . . . kT'^)
2l0) + (12r— 2) 63= (310) 63 = (111) e4 = (112)
210) 62 = (310) 63 = (111) f4 = (112)
210) + g{lSr-2) 62 = (310) + (1 2 r— 2) e3=(lll) e, = (ll2)
2l0) + (13r— 2) e2=(310) + (127--2)
111) + 2(12r— 2) + 2(13r— 2) e4=(ll2)
210) + (13r— 2) e2=(310) + (l2r— 2)
<'3=(lll) + 2(l3r — 2) e4 = (112)
210) + (12?— 2)— (13r— 2) 63 = (310) + (12 r— 2) e3 = (lll)
e, = (112)
e3 = (lll)+2(l3r— 2) e4 = (112)
63= (111) + 2 (12 r— 2)
63= (11 1) + 2 (13 r— 2)
e, = (112)
e, = (112)
210) + (l2r— 2) 6, = (310)
210) + (12r— 2) Co = (310)
210) e2 = (310)
When r = 4, 5, or 6. These cases may be found in XX.
200. Theorem. A Scheffers algebra of degree r — 1, which is not reducible,
must consist of two Study algebras, with one skew unit connecting them.^
201. Theorem. A Scheffers algebra of degree r — 2, which is not reducible,
must consist of
(A) Three Study algebras, E^, E., E^, with skew units (12), (23);
(B) One Study algebra, and one algebra of deficiency unity, with one
skew unit connecting them ;
(C) Two Study algebras, joined (1) by two skew units (12) (12), or
(2) joined by skew units (12), (21).
Theorem. A Peirce algebra whose degree is two, is determined as
2r— 1
202
follows :
for m
V8
we may take
Cj, Cj . . . . e,„ ^ £'1, such that EE^ := E^E =
The remaining units are such that
«m + 1 em + ^ = ^Ym + i.m+j, k Sk ^ = modulus, k=l ..
or in brief
E=E,^E„ E\ = 0, E,E, = E,E, = 0, E\<E„ ^i^j=-^j^i
m
> SOHErrsiis 8.
SCHEFFERS AND PEIRCE ALGEBRAS 65
One class of Peirce algebras of degree two, and order r, may be con-
structed from the algebras of degree two and order less than r, by adjoining
to the expressions for the algebra chosen for the base other terms as follows:
let the units of the base be e« . . . . e^. . . . . written with weight zero, say Cjo, e^]
then the adjoined unit (deleted unit) being e^-i = ?t,ii, we have for new units
^10 ^^ ^iO + ^« '^121 + ''(3 '''131 + • • • •
^jO^^^JO"^ ^J2 ^m M" ^13 ''-131 + • • • •
and Uij = — a_,( for all values of i, j.
The second and only other class involve units of forms ;i,ii + . . . . and are
given by
e^ = ^io + "gi' '^121 + + «2a ''-221 +
CjQ = «^0 + O^ ^121 + • + «^' ^221 +
^. = '^m — ^1 '^221 — ^2 ''-331 • • • • ^1, §2 • • ■ • = or 1
and afi^ = — a^' for all values ^ of i, j\ h.
203. Theorem. A Scheffers algebra of order r, degree two, consists of two
partial moduli ;i|,o + ^220 + • • • • + '^m,n.,o and ^m, + i.m, + i.o + + ''■r^,
and r — 2 skew units as follows"
'^Wi + 2 10. . . . '^rlO '^2 Wi + 1 '^3 nil + 1 ... . '^-mi m, + 1
204. The subject of the invariant equations of Peirce and Scheflfers
algebras is under consideration. Some particular cases are given later.
'Shaw 5. ^Scheffebs 3.
56 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
Vm. KRONECKER AND WEIERSTRASS ALGEBRAS.
1. KRONECKER ALGEBRAS.
205. Definition. A commutative algebra is one such that every pair of
numbers i^,, ^j in it, satisfy the equation :^
206. Theorem. An algebra is commutative when its units are commutative.
207. Theorem. The characteristic equation of a commutative algebra can
contain only linear factors, if the coordinates belong to the general scalar range.
208. Theorem. If the characteristic equation of a commutative algebra
whose coordinates are unrestricted has no multiple roots it is reducible to the
sura of r algebras each of one unit, its partial modulus. Such algebra is a
Weierstrass algebra.'
209. Theorem. If the characteristic equation of a commutative algebra
has p distinct multiple roots, it is reducible to the sum of p commutative
Peirce algebras. Such algebra is a Kronecker algebra.^
210. Theorem. The basis of a commutative Peirce algebra is a commuta-
tive algebra.
211. Theorem. A Kronecker algebra may contain nilpotents, a Weiek-
STRASS algebra can not contain nilpotents.* A Weierstrass algebra has
nilfactoriaLs.
212. Theorem. If the coefficients are restricted to a range, such as a field
or a domain of rationality, the algebra may not contain either nilfactorials or
nilpotents. Such cases occur in the algebras built from Abelian groups. This
case leads to the general theorem : If the equation of the algebra is reducible
in the given coordinate range, into p irreducible factors, the algebra is
reducible to the sum of ^j algebras and there are nilfactors. Each irreducible
factor belongs to one sub-algebra. If an algebra has an irreducible equation
in ^, the general number, such that the resolvent of this equation and its first
derivative as to ^ does not vanish, then all its numbers ma}- be brought to
the form
'(=b, e, + b, i + b,i~ + bsP+ .... + K_, i'-'
where i is a certain unit of the algebra, and i„ . . . . i^-i belong to the range.
If the resolvent vanishes for either a reducible or an irreducible equation,
there are uilpotent numbers in the algebra.^
'References for certain commutative algebras follow in the next article. On the general problem see
Stody 2; Fkobbnics2; Kroneckeu 1 ; Shaw 4.
»Sce references for §21.5, also KuoNF.cKEn 1.
> MOOBE 1. * KllONECKER 1. ' MOOBE 1 ; KnONBCKER 1.
KRONECKER AND WEIERSTRASS ALGEBRAS 57
213. Theorem. In canonical fornn the adjoined unil is of form
j= k K.i + 1 K.,+ ■■■■
« = 1 » = 2
Tiiere are as many terms of a given weight k as there are Vjasal units
with subscripts that appear in terms of weight k.
214. Theorem. The units of a commutative Peirce algebra may be taken
of the form.
^1 S2 • • • • Cm
where t, = .... jIa,; and where ^j"^', for i<; m, is linearly expressible in
terms of higher order.
2. WEIERSTRASS ALGEBRAS.
215. Definition. A Weierstrass algebra is a commutative algebra satisfying
the conditions ^'i^j=^jCi ^"^ whose degree equals its order,' and whose
coordinates are real.
216. Theorem. Numbers whose coefficient wi^ = are nilfactorial ("divisor
of zero"). The product of a nilfactor and any number is a nilfactor. There
are no nilpotents in the algebra."
217. Theorem. Tliere is at least one number (j, such that e^, g, g- . ■ ■ ■g'~^
are linearly independent. The latent equation resulting may be factored
into r linear factors, the imaginary factors occurring in conjugate pairs.
218. Theorem. A Weierstrass algebra is reducible to the sum of r' algebras
of the form
Xi A=i<-i ^iXj = ^ i,j=l.-- r' r = r'-\-r" i^J
and whose coordinates are scalars, which appear in conjugate forms if
imaginary (/■" is the number of algebras admitting imaginaries). Hence the
algebras may be taken to be of the form
Xi+Xi + l (Xi— X; + i)V— 1
with real coefficients ; or finally we may take the r' algebras as r' independent
ordinary complex algebras.
219. Theorem. Nilfiictors are numbers belonging to part only of the
partial algebras. If ^1,2. ...,i has coordinates in the first n algebras but not in
the other r' — ?i, ^„ + i....r ^^ coordinates only in the algebras from the
?i + 1-th to the ?-'-th, then
Sl .... n Sn + l....r' ^^
' Weierstrass 2 ; Scbwarz 1; DedekindI, 2; Bbrlott 1 ; Holder 1; Peterson 2; Hilbbrt 1;
Stolz 1 ; Chapman 3. The sections below are referred to Berloty.
'The preaeuce of nilpotents would loner the degree.
58 SYNOPSIS OF L.INEAK ASSOCIATIVE ALGEBRA
IX. ALGEBRAS WITH COEFFICIENTS IN ARBITRARY FIELDS.
REAL ALGEBRAS. DICKSON ALGEBRAS.
220. Definition. An algebra is said to belong to a certain field or domain
of rationality, when its coordinates are restricted to that field or domain. In
particular an algebra is real, when its coordinates are real numbers.^ The
term "finite" algebra is used also to mean algebras whose coordinates are in
an abstract (Gralois) field.
221. Theorem. The coefficients of the characteristic and the latent equa-
tions of an algebra are rational functions of the coordinates in the domain
^{x,y)} which is the domain of the coordinates and the constants" y.
222. Theorem. If new units are introduced by a transformation T' rational
in £l^, the new units are rational in £1^,; the hypercomplex domain ^ix,e) is
then identical with the hypercomplex domain n,^r_e')- Further, if £1^ contains
£ly, it also contains £L^,.
223. Theorem. If /S'. ^ is defined for any domain, then S . 1^,1% invariant
under any transformation of the units of the algebra and is rational in Hx.y
224. Theorem. In any domain there is an idempotent number or all
numbers are nilpotent.
225. Theorem. In a Peirce algebra every number ^ = ^o + ^i, where ^^ is
a multiple of the modulus, and ^j is a nilpoteut rational in Ilj.^. This
separation is possible in only one way. We may choose by a rational trans-
formation new units such that
e« = e^ eT' = i=\ ....r-\
The characteristic equation of ^ is F . C,=-t,'' [-^i^]'', where F , ^ is rationally
irreducible in £1^^.
226. Theorem. In any Scheffers algebra, we may choose by transforma-
tions rational in H^^, the units y; which are nilpotent such that
227. Definition. A real algebra may be in one of two classes, the real
algebras of the first class are such that their characteristic equations have no
imaginary roots for any value of ^, the general number; the second class are
such that their characteristic equations in ^ have pairs of conjugate imaginaries.'
228. Theorem. Every real quadrate is, if in the fii'st class, of the form
6'^ ^" '^ (1 ) <-,;■ e,i = ^jke,i iz=i.... J)
If of the second class, it is of order 4j/, and is the product of Q and an
algebra of the first class (l).
'Dickson 5; Tabek 4. Hamilton restricted Quaternions to real quaternions, calling quaternions with
complex coordinates, biquaternions.
'Taber 4. The succeeding sections are referred to Taber 4. This paper contains otber tbeorems.
3CARTAN 2. This reference applies to §§228-232.
ALGEBRAS WITH COEFFICIENTS IN ARBITRARY FIELDS 59
The algebra Q is Quaternions in the Hamiltonian form
e^, h 3, k, ij = —ji = /c, etc.
229. Theorem. Every real Dedekind algebra is the sum of algebras,
each of which is of one of the following three types :
(1) Real quadrates of first class;
(2) Real quadrates of second class ;
(3) The product of a quadrate of first class and the algebra eg, e,, where
230. Theorem. Every real Scheffers algebra of the second class is derivable
from one of the first class by considering that each partial modulus belonging
to a complex root of the characteristic equation will furnish two units for the
derived algebra, say
gj =: Xj + Xj ^2 = ('f 1 — ^2) *^ — 1
That is, the direct sub-algebra consists of direct nil-potent units and of the
sum of algebras of the forms
Co or eo,ej (^1 = — Cq)
All other units are chosen to correspond ; thus r^^^ furnishes two units, rj^ and
>7.^, corresponding to x^i *^ — 1 ^i'
231. Theorem. A Cartan real algebra is primary, and has a Dedekind
sub-algebra according to §229, the other units conforming to this sub-algebra
in character, and giving multiplication constants y which are real ; or it is
secondary, and has a Dedekind sub-algebra consisting of the algebras in §229
multiplied by real quadrates of the first class, the other units conforming as
usual.
232. Theorem. Every real irreducible (in realm of real numbers) com-
mutative algebra is of the types of §230. It is a Peirce algebra then, the
modulus being irreducible; or else it has two partial moduli which give an
elementary Weierstrass algebra, and hence are irreducible in the domain of
real numbers.
233. Theorem. The only real algebras in which division is unambiguous
division (in domain of reals) are (1) real numbers; (2) the algebra of complex
numbers eg = ^o = — ^ ^j := 60^1 = e^e^ (3) real quaternions.^
234. Definition. A. Dickson algebra is one whose coordinates are in an
abstract field.
235. Theorem. The only Dickson algebras (associative) which admit of
division are those whose coordinates are in the Galois (abstract) field, and
whose qualitative units are real quaternions or sub-algebras of quatemions.-
■Frobenius 1; C. S. Pkirce 4; Cartan 3; Grissemakn 1.
s Weddebbcrn 4. See also Dickson 6 and 7.
60 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
X. NUMBER THEORY OF ALGEBRAS.
236. Definition. The number theory of an algebra is the theory of domains
of numbers belonging to that algebra. Algebras usually do not admit of
division, unambiguously, hence the term domain is taken here to mean an
ensemble of numbers such that the addition, subtraction, or multiplication of
any of the ensemble give a result belonging to the ensemble. The first case
which has been studied is that of quaternions, which admits division.'
237. Definitions. An infinite system of quaternions is a corpus if in this
system addition, subtraction, multiplication, and division (except by 0) are
determinate uniquely.
A permutation of the corpus is given by \f^^\ if through the application
of this substitution, every equation between quaternions in the corpus
remains an equation. Hence
f{a + b) = f{a)+f{b) f(ab)=f{a).f{b)
If n is the corpus of all quaternions we have the substitutions
(2) /(a) = flo =t «i K ± «2 h =^ ^^S ") i^> y is a permutation of the
indices 1, 2, 3.
238. Theorem. If R is the corpus of rational quaternions, then a is
rational when Ug, aj, a,, a^ are rational. The permutations for the rational
corpus ai*e q ( ) q"^, and (a, /?, y).
239. Definitions. If p = 5 (1 + i'l + ig + is), and (? = A'op + ^i rj + A-, ij
+ A-gig, where Ic^, Ici, ho, k^ are any integers, q is said to belong to the integral
domain J.
If the cooi Jinates of q, h A-q, J A^ + A-j, i A.-^ + k^, i A;o + A;3 are integers,
q belongs to the sub-domain Jo*
An integral quaternion is one which belongs to /.
An integral quaternion a is pre-divisible (^post-divisible) by h if a z= be
(a = cb) for some integral quaternion c.
If e and £~' are both integral, e is a umV. It follows tliat
N{E)={TEf= 1
There are 24 units:
. ± 1 ± i, zh i» ± t
f=±l, ±ii, ± *2> =t *3, -^
240. Theorem. If a = «c, then a^=^ c^v only if y = re or ?v, where
^ = 1 -}- i|, and r is any real integer.
'HuRWiTZ 1. Of. LirscniTZ 2. The first reference applies to all sections following to §257.
NUMBER THBORY OF ALGEBRAS 61
241. Theorem. I fa and h iiru intcgriii (|iiiit,eriiioMK, // :^ 0, we can find
q, f, Y, , f, so that
a = qb-\-c a = hq, + 6-j N{c) < N{h) N{r.,) < N{h)
242. Theorem. Every two integral (|uuternioMs a and />, which are not
both zero, have a highest common post-factor of the form
d = (ja + hb ((J J hj integers)
and a highest common pre-factor of the form
di = or/, 4- A//, (^,, /<, , integers)
243. Theorem. The quaternions 0, I, (j, f form a complete system of
residues modulo ^.
244. Theorem. A quaternion belongs to J^ if it is congruent to zero or 1
(mod 0-
245. Theorem, li N . a{= N . K a) \s divisible by 2', then a = '('h where
h has an odd norm.
246. Theorem. The following quaternions form a complete system of
residues, modulo 2 :
1, h, H, «3 '-^ 0, 1 + ij, 1 + I.,, 1 + i,
247. Definition. A primari/ quaternion is one which is congruent to either
1 or 1 + 2p (mod 2^). Every primary quaternion belongs to Jq.
Two integral quaternions are ^)re- (post-) associated, if they differ only
by a pre- (post-) factor which is a unit.
248. Theorem, Of the 24 quaternions associated to an odd quaternion i,
only one is primary.
249. Theorem. The product of two primary quaternions is primary.
250. Theorem. If b is primary then when
(1) b=l (mod 2^), K. b is primary,
(2) b = l+2p (mod 2^), —K. b is primary.
251. Theorem. If ??i is a positive odd number, the m* quaternions
?o + ?i h + q-z h + qsh (qo, qu qi,q^ — ^, i, 2 . . . . m — 1)
form a complete system of residues modulo m.
These quaternions are holoedrically isomorphic with the linear homo-
geneous integral binary substitutions :
ccj = aa-'i + /3a-2 xl = yxi + 6x0 (mod m)
N{aS — (3y) = N.q {mod m)
252. Theorem. The number of solutions of N{q) = (mod m), q being
prime to m, is vi^U (l — ., K^ H )
62 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
The number of solutions of N {g)=l (mod m) is m^Tlfl 2}'
These form a group G,n which is holoedrically isomorphic to the group of the
linear homogeneous binary unimodular integral substitutions, modulo m.
253. Definition. 7t is a ^n'me quaternion when its norm is prime.
254. Theorem. There are ^j + 1 primary prime quaternions whose norm
equals the odd prime p.
255. Theorem. If N. c=p''q'' .... then c = Ti^Tta .... ni,Xi .... x^. ... . where
Ttj, X, . . . . are primary prime quaternions of norms p, q, etc.
256. Theorem. If m is any odd number, there are ^ (m) = 2 . (^ (sum of the
divisors of m) primary quaternions whose norm equals m.
257. Theorem. The integral substitutions of positive determinant which
transform xl + sq -\- x:^ -\- xl into a multiple of itself are given by the equations
1/ = ax(3 y = — a . Kx . /B
where a, (3 are any two integral quaternions which satisfy the conditions
a/? = or 1 (mod^.
258. Definition. The general number theory of quadi-ates has been studied
recently by Do Pasquier.^ A number in a quadrate algebra he calls a feftarion.
It is practically a (square) matrix or a linear homogeneous substitution. An
infinite system of tettarions is a corpus, if when a and ^ belong to the system,
a ± ^, a . ^, (3 . a, a : (3, (S~^.a belong equally to the system. A substitution
of a tettarion t = /(t) for a tettarion t is indicated by [t, /(t)]. K permuta-
tion is a substitution such that when a is derived from n tettarions aj. . • -a,, by
any set of rational operations, so that a =-fi(ai. . • .a„), then /(a) = a is
derived from aj. . . .a„ by the same set of rational operations, so that
a = .B (cci . . . . a„)
259. Theorem. The substitution [a, /"(a)] is a permutation of the corpus
\K\, if the tettarions /(a) do not all vanish, and if
/(a + /3) =/(a) +/(/?) f{a[3) = /(a) /(^)
for any two tettarions a, (3 in \K\. The tettarions /(a) also constitute a
cori)us.
260. Definition. An inversion of the corpus is a substitution such that not
all /(t) are zero, and also for any two tettarions a, (3 we have
f{a + (3) = f{a) + /(/?) /(a/3) = /{(3) /(a)
[t, t] is an inversion, where t is the transpose of t. If [«,/(«)] is the most
general substitution of the corpus, [a, /(a)] is the most general inversion.
261. Definition. Two permutations of the form
[a, /(a)] and [a, q . /(a) . q-'}
where q is any tettarion which has no zero-roots, are said to be equivalent. All
equivalent permutations constitute a class.
> Do Pasquiku 1. This reference appHes to SS 858-297.
NUMBRR THEORY OF ALGEBRAS g3
262. Theorem. The .siib.stitution t,,t ' is a perniutation of the corpun o
of all tettario.is of order .v; where t is a tettarion such that Nit)^: N U)
being the .s-th or last scalar coefficient in the characteristic equation of 't The
coefficient N{t) is called the norm o{ t.
263. Definitions. A tettarion is m^ioW if all its s^ coordinates are rational
All rational teltarions form a corp„s \R\. All tettarions whose coordinates
belong to a given domain of rationality constitute likewise a corpus A
rational tettanon is integral if all its coordinates are rational integers.
The integral tettarion a is pre- (or post-) divisible by the tettarion 3 if
an integral tettarion y can be found such that a = f3y (or a = y^) A unit
tettarion . is an integral tettarion which is pre- (post-) divisible into every
integral tettarion. When Nie) = + 1 we call . a proper unit-tettarion ; when
-^ (^) 1 we call e an improper unit-tettarion.
264. Theorem. Let a,. =h + e„ where h is the modnhis of the quadrate
that IS, IS scalar unity, and e^. is one of the s^ units defining the quadrate ; and
^^^ ^ ~ 5 *'' ''' ^^ *"^ integral tettarion ; then among the tettarions
'^ ~o.ur (a;=l, 2....)
there is always one such that a certain pre-assigned coordinate, say tif is not
negative, and is less than the absolute magnitude of any other coordinate of r
of the form t,j {k=l.... s, k zfz i), provided t,j :^ 0.
265. Definition. A tettarion yf^e, in which all coordinates for which
^ > A; (or * < ^ vanish, ,s said to be pre- {post-) reduced. They constitute a
sub-corpus^ lettarions of the form y,t,,e, are both pre-reduced and post-
reduced The components ^,(. = 1 .....) in a reduced tettarionr vanish only
when T has zero-roots. ^
266. Theorem. If t is any integral tettarion, a proper unit-tettarion e
may be found such that . . r (or r . ^) is a pre- (post-) reduced tettarion, in
which, of all the coordinates <«, at most only t^ can be negative. This co-
ordinate is negative only if iV(T) < 0.
If r is any integral tettarion, we may find a pair of proper unit-tettarions
e, and e, such that e, re, is of the form 2 cZ, .„ (i = 1 . . . . ,), and among the
coordinates at most only <, is negative, and cZ., is divisible by d- , ._, The
coordinate d,, is negative only when A^(t) is negative.' ' '
If a = ei /?f2, a and /? are said to be equivalent.
267. Theorem. Every proper unit-tettarion e is expressible in an infinite
number of ways as the product of integral powers of at most three unit-
tettarions. These three may be
tty = A -f- ey
^ij = ^ e,^ + Cij - eji {k=i .... s, k 4: i, k :^j)
7 = ^21 + % + . . . . + e,,_i — e„
'Cf. Kkoneckbk: CrelleW, 135-136; Bachman: ZahlenthtorU IV T^il, 29i.
64 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
268. Theorem. Every integral tettarion r is equivalent to a tettarion of
the form S tu e^. The coordinates less t are the shear factors of the character-
istic equation of T. The norm of t, N'{r), is the pr9duct of these coordinates.
Two tettarions are equivalent when they have the same shear factors and the
same nullity.
269. Theorem. In order that a, be a factor of t = aj . . . . a^ . . . . a, it is
necessary and sutficient that the nullity of ai be not higher than that of t, and
that each shear factor of a,, or combination of shear factors, be divisible into
the corresponding shear factors of t. If an integral tettarion t is a pi'oduct
of others, then every combination of shear factors of t is divisible by the
corresponding combination of shear factors of any one of these others.
270. Definition. Two tettarions t and er are called pre-associated . The
association is, proper or impropter according as N{£) = + 1 or — 1. Associated
tettarions form a class. The simplest representative of a class will be called
a primary tettarion.
A pre-primary tettarion p = ^Pijeu satisfies the following conditions :
Pij = ^ i>J Pn = ^ and Otpij<Pji
for all i<^j and if pjj -^ 0; i and j have values 1 . . . . s subject to the con-
ditions stated.
271. Theorem. A primary tettarion cannot have a negative norm. Pri-
mary tettarions with zero-roots are infinite in number, but the number of
primary tettarions of a given norm m :|: is finite.
272. Theorem. If wi = n j:5f , where p^ is a prime number, and if xijn)
i
is the number of distinct pre-primary tettarions of norm m, then
xM = xipf) x(pf') ■■■■xipf)
_ (ff« + I-l)(p"+2_i) ....(^,a-f.-l_l)
nP)- (^,_l)Q/_l)....(^s-l_l)
273. Theorem. If r is any integral tettarion and m is any integer (rf: 0)
then an integral tettarion a may be found such that r:=ma or else T=^7nG + a
wherein a is an integral tettarion and ■< | N{a) ] <! | m' | .
274. Tlieorem. If a and 8 are two integral tettarions of which 8 has not
zero-roots, then two integral tettarions j3 and y may always be determined
such that either
a = (3h and y = or a = {38 + y and < | A^(y) |< | A^(5) |
By this theorem a highest common pre- (post-) divisor ma}' be found by
the Euclidean method for any two integral tettarions.
275. Definition. An infinite system of tettarions which do not all vanish
is a pre- (jjost-) ideal if when Tj and Tj belong to the system, ^Tj, Tj ± Tj also
belong to the system, where y is any integral tettarion.
NUMBER THEORY OF ALGEBRAS 66
276. Theorem. If rj . . . . t„ are iiitej^ral lettarions, which do not all
vani.sii, tljcii tlio totality of tettarions 7, t, + .... y„r„ wliere yj . . . . y„ run
independently throiij^h the range of all integral tettarions, is a posl-ideal.
The tettarions t, .... t„ are said to form the basis. An ideal with one
tettarion in its ba.sis is a principul ideal. It is designated hy {ry) or (yr).
Two ideals are equal if they contain the same tettarions.
277. Theorem. Pre- (post-) associated tettarions generate the same prin-
cipal post- (pro-) ideal. If two integral tettarions without zero-roots generate
the same principal post- (pre-) ideal they are pre- (post-) associated.
278. Theorem. Every pre- (post-) ideal generated by rational integral
tettarions which do not all have zero-roots is a principal ideal.
279. Definition. Nul-idenls contain only tettarions with zero-roots.
280. Theorem. An ideal which is both pre-ideal and post-ideal cannot be
a nul-ideal.
281. Theorem. Every n given integral tettarions a, (3 . . ■ ■ (i which do not
all have zero-roots possess a highest common divisor h which is expressible in
the form
5 = a^i + /?^2 + • • • ■ + i"^.i or h =■ hi(x + ^o^ +....-{• Sn(i
wherein 5j(i= 1 .... n) are definite integral tettarions. Every pre- (post-)
divisor of 5 is a common divisor of a .... jU and conversely. Moreover S is
determined to a factor which is a post- (pre-) unit-tettarion.
282. Theorem. If rx and /3 have no common pre- (post-) factor, then two
tettarions y, 6 may be found such that
ay + ^6=1 or ya + 6(3 = 1
If a and (3 have a common factor these equations cannot be solved. If a and
/? have a common divisor which is not a unit-tettarion then N{a) and N{l3)
have a common divisor other than unity. The converse of this theorem and
the theorem imply each other if one of the tettarions is real, that is, of the
s
form a 2 e^j.
<=i
283. Definition. An integral tettarion which is not a unit-tettarion nor has
zero-roots is prime if it cannot be expressed as the product of two integral
tettarions neither of which is a unit tettarion.
284. Theorem. The necessary and suflBcient condition that 7t is a prime
tettarion is that its norm N [n) is a rational prime number. There are
»' — 1 .
— ^~: different primary primes of norm p.
285. Theorem. If h is an integral tettarion of the form 2 </,, e,j and if
N {h) =■ p'^ q'' . . . .t", where p, q. . ■ -t are distinct primes, not including unity,
then S can be factored into the form
,S = Ttj .... 7t„ Xi .... Xb .... Ti .. . T„
66 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
where n^. . ■ -rCa are primary prime tettarions of norm p, xi- . ■ -x^ are primary
prime tettarions of norm q,. . . .r^. . . -Tn are primary prime tettarions of norm
t, all being of the form
Xpueu 2<«ete
286. Definition. An integral tettarion is primitive if its coordinates have no
common divisor other than unity. It is primitive to an integer m when its
coordinates are all prime to m.
Every primary tettarion is also primitive.
287. Theorem. Let y be a primitive integral tettarion and
where 2?, q- ■ • -t are the prime factors o^ N{y). Then y can be decomposed in
only one way into the form
where e is a unit tettarion, and Tti- . ■ -Tia, are prime tettarions of norm p,
Xi- . • .Xa„ are prime tettarions of norm 5, • • • • Tj . . • . T„„ are prime tettarions of
norm t. The product of each I successive factors is primary, where
Z= !....«. (i= l....n)
288. Definition. If aj, a^. . . -a^ are s integral tettarions of equal norms
N{ai) = iV^(a2) . . . . ^ ^{o-s), and if ai a, . . . . a^ = iV(ai), then these tettarions
are semi-conjugate.
289. Theorem. A product of any number of prime tettarions of forms
Sf^iiCu is a primitive tettarion of the same form if among the factors no s of
them are semi-conjugate.
290. Theorem. A product of primary prime tettarions ni...-7t„, where
N(^'7ti)=pi (i = I . . . .71) pi being distinct primes, is always a primitive
tettarion.
291. Definition. Two given tettarions a and (i are pre- [post-) congruent to
a moduluti y, if their difference a — |S is pre- (post-) divisible by y. This con-
gruence is indicated by
a = /? (mod y, pre)
or
a — ^ (mod y, post)
There is then an integral tettarion ^ such that
a — (3 = y^ or '(y
292. Theorem. If a and /? are pre- (post-) congruent modulo y, they are
also pre- (post-) congruent for any tettarion post- (pre-) associated with y, as
modulus.
NUMBER THEORY OF ALGEBRAS 67
293. Theorems. If rx = 'r /^ = t then a = /? (mod y)
11" a = T ii = a then arh/S^Tia (mod yj
If a = /3 then ra = r^ (mod y, post)
If a = /? then ar = (3r (mod y, pre)
If7<^ = i^y a = /3 then a^S/^i^ (mod y, post)
If rx E^ /:; = (5 y(3 = ^y then e« = ^/5 (mod y, post)
294. Definition. Two tettarions a, (3 are congruent as to a rational integer
in ipO, when a — /? is divisible by tti, indicated by a = ^ (mod w). In this
case for each pair of coordinates we have afj = bij (mod m). A complete
system of residues consists of wi"' tettarions, obtained by setting each coordinate
independently equal to each one of a complete set of residues modulo m.
295. Theorems. A tettarion congruence modulo m, a rational integer can
be divided by an integral tettarion ^, without altering the modulus only if
iV(^) is prime to m.
A sufficient condition for the solubility of the congruence a^ = /5 (mod m)
by an integral tettarion ^ is that N{a) is prime to m. If this condition is ful-
filled the congruence possesses one and only one solution ^ (mod m), namely,
^ = r — ^/3 (mod m), where r is a root of r . N{a) = 1 (mod m).
296. Definitions. A tettarion with zero roots and nullity s — r is pseudo-
r
real if it is of the form c?jj "S, e^^, r <is. A tettarion with zero-roots, of the form
2 t^jCy where t^^ =0 for/ = r + 1 • • • -s, is singular or non-singular according
as its rank is less than or equal to r. The product of the first r latent roots
of a tettarion of this kind is called its pseudo-norm N'. A tettarion of the type
2 <ii Cji is never singular. When a tettarion is singular its pseudo-norm is zero.
297. Theorem. If a and fi are two integral tettarions in which coordinates
of the form (,j =. for J = r -{• 1 . . . . s, and if a is pre-reduced, u is pseudo-
real, and ^ 0, then two tettarions of the same type ^ and >;, may always be
found such that either
a = (I .^ )7 =
where the pseudo-norm of a satisfies the conditions
0<\N'{a)\<\N'{f^)\ = \{m,,y\
If r and (3 are two integral pre-reduced tettarions of the same type as
a, fi above, and ^ is non-singular, then there are always two other pre-reduced
tettarions of this type ^ and y; such that
r = 5,5 )7 =
^^ r = q^ + y! and < | iV^' (>?) [< 1 iV' (,5) |
Every non-singular post-ideal based on tettarions of this type is a princi-
pal ideal.
68 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
XI. FUNCTION THEORY OF ALGEBRAS.
298. Definition. In §58, chapter II, we have for any analytic function of ^,
This definition gives a complete theory, if the roots may be treated as known.
Other definitions are given below.^
CO
299. Definition. 2 a, ^'' defines an analytic function of ^, if the roots of the
1
CO
characteristic equation of ^ converge in the circle defined by 2 a^z", where sis
1
an ordinary complex number.
Sa^^" defines a function of ^, if ^~^ exists, and if the roots of the char-
CO
CO CO
acteristic equation of ^ converge in the circles^ of 2 a^z" and 2 a_^2"'^
1 1
300. Definition. Let
f=^eji{x^ a:,) i—1 r
and let
(ia; = 2 Cj dxi
Then
df = y^ 1^ . dx,e, = f'.dx = dz.<f
If tt' =^1, then /is an analytic function^ v/hen/' .y ^ y .'/.
301. Theorem. The algebra must contain for every number m, a number
V such that ?/_y = yv for every y.
302. Theorem. In a commutative algebra the necessary and sufficient
condition of analytic functions is
303. Theorem. The derivative of an analytic function is an analytic func-
tion. If two analytic functions have the same derivative, they differ only by
a constant.
304. Theorem. An analytic function is a differential coefficient, only when
the algebra is associative, distributive, and counnutative.
Ill
305. Theorem. Km and v are analytic, uv and are analytic.
' Fbobbnius 1 ; BuoHHEtMT; Sylvester 4 ; Tabbk C, 7. 'Wetr 7.
'ScHEFFEKS 8. Applies to §§300-306.
FtJNCTION THEORY OF ALGEBRAS 69
306. Theorem. ir/(px) = f(x), tlicn /(y) is a constant.
307. Theorem. For a Weierstuass commutative algebra, let «,, h, he in
the i-th elemcutary algebra, and
a = 2 cit b =z2hi
Then a ± i = 2 (a, ± b,) ab = Xa, hi " = 2 ^' , if b is not a nilfactor.
If hi = 0, for t = 1 . . . . ij, 6t ^ i> ii
and if a^ = 0, for t := 1 . . . . ij
then -^ = 2 -T- i > M
In any other case the division of a by Z/ gives an infinity}
308. Theorem. The sum, difference, product, and quotient of two poly-
nomials is formed as in ordinary algebra.
309. Theorem. The number of solutions of an algebraic equation of degree
p is N-=. p^', when each elementary algebra is of order two.
If ri of the elementary algebras are of order one, and ?•• — r^ of order
two, N=-p\
In any case the number of infinities and roots is ^/. The nun)ber of roots
is infinite if, and only if, the coefficients are inultijiles of the same niliaclor.-
310. Theorem. A polynomial i^(^) can not vanish for every value of ^
unless its coefficients all vanish.
Two polynomials equal to each other for every value of ^, must have
the coefficients of like powers of ^ equal.
311. Theorem. If an algebraic polynomial i^(^) is divided by ^ — ^', ^'
being a root, the degree is reduced to {p — 1) and/j*"" — {p — l)'"" roots have
been removed.
In ordinary complex algebra r= 2, jj**" — {p — l)'"" = 1.
312. Theorem. If two polynomials have a common root, ^', they have a
common divisor ^ — t,'.
313. Theorem. If F{j^) is differentiated as if ^ were an ordinary quantity,
giving F' {Q, then the necessary and sufficient condition that there is a system
of roots of i^(^), having just p equal roots, is that F' (^ has at least one
system of roots of which p — 1 are this same equal root, and that no system of
roots of F'{(^) has this root more than p — 1 times. F{^) and F' {^) have
therefore the common divisor (^ — ^'Y'^'
314. Theorem. It is not always possible to break u-p ,, ^L into partial frac-
tions.
'Bbrlott 1. Applies to §§307-315. 'Weierstbass 2.
6
70 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
315. Theorem. If ^ is considered to be written in the form ^ = 22; Xj,
where i = 1 . . . . rj, and Zj is any real or complex number, the whole theory of
functions of a comj^lex variable may be extended to numbers which are not
nilfactors. If there are nilfactors, meromorphic functions must be treated
specially. We have
316. The treatment of quaternion and biquaternion differentials, integrals,
and functions may be found in the treatises on these subjects and references
there given ; references are also given at the end of this memoir. The general
principles of such forms may easily be extended to any algebra. Differentia-
tion and integration along a line, over a surface, etc., may also be found in
the appropriate treatises.
The problem of extending monogeneity to functions of numbers in
quadrate algebras has been handled recently by Autonne.^ His results are
as follows :
Let ^ be any number in an algebra, and let H be a number wliose
coordinates are functions of those of ^. The index of monoyeneity N is the
N
minimum number of terms necessary to write cZH in the form 2 Gi . dt, . r^,.
i = \
'9
wherein Ci and Tj are functions of ^. If we write v = 2 e{ >, - , we have in all
i = \ VX^
cases c^H^ /• (^^V • H = T(t7$). The Jacobian of the coordinates of E is
then 7n,.(T).
1 — r
If now we put T= 2 u^j K^i, where K^^i = e,, Q e, , we may find the
H
scalars w^i uniquely if the algebra is a quadrate." For, indicating quadrate
units by a double sufiBx, and writing n" = r,
ijkl
and if we operate on Cj^. and take /. e^ () over the result,
l....r
If we put 'P = 2 ■?% . Ci /e, (), or in the case of a quadrate,
hi
\....n
* = 2 w^j„i . ey lea
ijkl
then the rank, that is, ?i — v, where r is the nullity of ip, is the index of
monogeneity, N. N is invariant for a change of basis.
The transverse of 'P corresponds to interchanging cr, and r^. For
and /e^y Ten = w,
„ 1 n 1 — n
* = 2 lea ^ejk ■ ^ki l^a — 2 le^j Ten . e^ /e„
ijkl ijkl
' AUTOMNB 5, 6. ' HACSDOBFr 1.
FUNCTION THEORY OP ALGEBRAS 7 j
Let P = 2 Ci me, 0, ti.o ^^ forming a ^-pair. Then P = P, and we
i ^1
have
?r = :S.w,,^e,me,c,{)e,^)
kl
= 2 rvki 2 c, hi . /^ (e< Cj c, e^t
PT = 2«;«2e,/e,()./^(e,eie,e,0
Hence PT = T P = PT if ip = 'P, and conversely. Again PT^PH/y,
therefore ifPT = fP we have
PH/v = y/P3
Operating on cZ^, we Iiave
c? . PH = ^Td^PB — \7lBdP^
Hence if y; = P^, /&?>? is an exact differential. Thus if V is self-transverse,
/E(/>7 is an exact differential and conversely.
When N= 1, we have H in one of the four following types :
I. B = K^A + M (K, A, M, constant)
II. H = 2 Z (^e,i) fi,i (Z arbitrary)
i^ 1
III. E = 2 ^n Z (fii,^) (Z arbitrary)
IV. B="i\ufr,{t)p,{t)dt
tj
< = t|' [/. a («'u ^) . ■ . • I. a (e,„i^)], and i^, >7,-, p^ are arbitrary scalar
functions oi t; a is any constant number.
72 SYNOPSIS OP LINEAR ASSOCIATIVE ALGEBRA
Xn. GROUP THEORY OF ALGEBRAS.
317. This part of the subject is practically undeveloped, although certain
results in groups are at once transferable to algebras. A considerable body
of theorems may thus be got together, esjDecially for the quadrates. For
example, the groups of binary linear homogeneous substitutions lead at once
to quaternion gi'oups, ternary linear homogeneous substitutions to nonion
groups, etc. It is to be hoped that this branch may be soon completed.^
318. Definition. A group of quaternions is a set of quaternions Ji • • • ■ ?„ ,
such that
qTi = l 9[iqj = qk i,j=l....n
TTii is a positive integer, and k has any value 1, 2....n. The quaternions
give real, complex, or congruence groups according as the coordinates are real,
complex, or in an abstract field.
319. Theorem. To every quaternion q'=-w-\xi -\- yj + zlc corresponds
the linear homogeneous substitution
/w -\- z *^ — 1 — 2/ -j- a; V — 1\
and conversely. The determinant of the substitution is I'q. To the jiroduct
of two quaternions q, r, corresponds the product of the substitutions.
320. Theorem. To every group of binary linear homogeneous substitutions
corresponds a quaternion group, and conversely. To every group of binary
linear fractional unimodular substitutions corresponds a group of quaternions
multiply isomorphic with it, and to every quaternion group corresponds a
group of binary linear fractional unimodular substitutions, the latter not
alv?ays distinct for diflerent quaternion groups.
321. Theorem. To every quaternion of tensor Tq corresponds a Gaussian
operator Tq.q Oq^^ = Oq, and conversely.
If 2 . r = s, then Gg . (?,. ^ G^-
Hence groups of these Gaussian operators are isomorphic with quaternion
groups, and conversely, but tlie isomorphism is not one-to-one.
322. Theorem. To every unit quaternion q, there corresponds a rotator
^,^ = 2 () g-i, and conversely, the same rotator corresponding to more than
one quaternion.
Likewise a reflector Ii,j=: — qQ q~^, and conversely.
Further, for any fixed quaternion a admitting of a reciprocal, there cor-
responds the a-transverse of q,
T\f = aqa-^
' Cf. Ladbbnt 8, 4.
GUOUP THEORY OF ALGEBRAS 73
Thus if qr = 8,
R,^ . n, - R, R., .11, = — R, Tf^ . T\ = T'i")
Thus to every group of quaternions g-i. . . .(/„, corresponds the rotator group
R^^ .... R^^ ; tlie reflector group ± R,^^ , ± R,^„ . . . . ± R,^^ ; and the transverse
groups 7',*,''' .... T;il. If « = 1, the transverse group is the group of conju-
gates ; and if /Set =: 0, we Iiave a group of transverses in the matrix sense.
323. Theorem. If weconsider that gand — q are to be equivalent, </ = — q,
then tlie rotation groups give tlie quaternion groups as follows:
To C„ corresponds h"n , r = 1 . . . n
Z>„ corresponds lcn,i, r = 1 . . . -n
T corresponds 1, i,j, h, (1 ± irby ± Ic)
corresponds 1, i^, fi, ki, ^(1 ± i±j±i /•)
hx/2{i±J), iv/2(y±7.;), hK/2{fc±i) r=l, 2, 3
7 2'!//.. „, »„ov 7 2h"
_ 17 2/1 ."h k 6 (i -{- 2/e cos 72 ) « 6
/ corresponds k = , jk 5 , -A^^_p.^^^._=-
A;f y(l +J/^cos 72°) 7cT ^^ ^^, ^„ ^ ^ .
V 1+4008^72° ' '
324. Theorem. To the extended polyhedral groups correspond the follow-
ing five quaternion groups :
in
To Cr corresponds the group hr , of order 7; {k any unit vector,
n= 1 .... r).
To Dl corresponds the group 7c^ i'\ of order 4r, {Sik= 0, i? ■=■ — 1,
n=: 1 . . . .r; 7t = 1 . . . .4).
To T corresponds the group of order 24: ±1, ± t, d= y, ± ^,
i(dz 1 =b t ±y ± /.•).
To 0' corresponds the group of order 48 : ± 1, ± i, iy, rfc h^
^ (± 1 ± ?• ± y ± A) ^V2 (± 1 d= i) i V2 (± 1 ± y)
i V2 ( ± 1 ± /.•) i>/2 (± i ifc y) i V2 (it y ± z,-)
i ^2 ( ± A; db i)
To /' corresponds the group of order' 120 : ± k\ , ±:jkT,
it k''s (i + (o/r) ks it k'i {i + ak) jkT
V 1 + 0)^ Vl -\- (^
• ■5 n + s = it l,it 2(mod5) u = 2cos 72° = J (— 1 -f V5)
1 Of. Stbimobah 3.
where
74 SYNOPSIS OF LINEAR ASSOCIATIVE AIXJEBKA
325. Theorem. Combinations of rotations and reflections give the poly-
hedral and the crystallographic groups. Thus we have correspondences :
C, =kT{)Jc--: n=l.... r
Dr=kr{)k-J i{)i-' n=l.... r
T =1 h{)h-' i{)i-' j{)r' {\±i±j±ic){){\±i±j±k)-'
=1 i{)i-^ JOf^ k{)k~^ (1 d=iifcy± ^)() (1 ±i±yd=A:)-'
(1 ± 00(1 ± i)-' (1 ±y)()(i ±y)-^ (1 ± k) (1 ± k)-'
[i ± j) ( i ±y)-i (y ± /.•) (y ± k)-' {k .4= i) o (/.: ± i)-
/ = Z;^ k-i J ()y-i (i + 2 cos 7 2° /;•)() (i + 2 cos 7 2° k)'^
2r
. r
. 2/-
and their combinations.
6^; z= [— k'^ {) 7c-''-^Y h =
(_!'J =z k'v /.-.^T — /i- A; ' and couibination.s /* =
C'J' =^ k~r {) 7c~ ~r — i{)i~^ find combinations h =
D'r = [ — kH' 0^^'^ ] ^Oi"^ and combinations A =:
Z>,'.' = ^T &~ 7 — k\)k~^ i()t~' and combinations A =
Z)r"= k~r {)k~~f — a()a~* iOi"^ oil^ ii^nd combinations Ji =
rpi rp m
T" = T [ — (* — y) (i — y)"'] a.nd combinations
0' = —
/' =/ — /
326. Theorem. If /S'. e = f, t'" =1 ; then the product of each group in §324
into the cyclic group of e, gives a group of quaternions.
327. Groups of quaternions whose coordinates are in an abstract field,
remain to be investigated.
328. Theorem. The continuous groups^ of quaternions are as follows:
(1) All quaternions,
(2) All unit quaternions.
(3) Quaternions of the form w •\- xi -\- y% \ Si'^=^Q =-^-.
(4) Quaternions of the form w-\-y^; (S'may=:y+ "^ — 1 k).
(5) Quaternions on the same axis, w + xi.
(6) Scalars, rv.
(7) The quaternions f +i 5(1+ ^/^^l i) + <° J (1 — -v^I^l *) + y%,
t arbitrary.
(8) The quaternions e' + te'-'^.
(9) The quaternions 1+^3^.
(10) The quaternions f + 4(1 + V^^l i) + f \{\ — ^/^^i).
1 ScnEFFBRS 7.
GENERAL THEORY OF ALGEBRA 75
Xin. GENERAL THEORY OF ALGEBRA.
329. While this memoir is particularly concerned with associative linear
alyebra, it is nevertheless necessary, in order to place the subject in its proper
perspective, to give a brief account of what is here called, for lack of a better
title, the general theory of algebra.
The foundations of mathematics consist of two classes of things— the
elements out of which are built the structures of mathematics, and the 2mjcesses
by which they are built. The primary question for the logician is: What are
the primordial elements of mathematics? He proceeds to reduce these to
so-called logical constants :^ im^iUcation, relatiun of a term to its class, notion of
such that, notion of relation, and such further notions as are involved in formal
nnplication, Viz. , propusitional function, class, denoting, and any or every term.
To the mathematician these elements do not convey much information as to
the processes of mathematics. The life of mathematics is the derivation of
one thing from others, the transition from data to things that follow according
to given processes of transition.
For example, consider the notions 3, 4, 7. We may say that we have
here a case of correspondence, namely to the two notions 3, 4 corresponds the
notion 7. But by a different kind of correspondence, to 3, 4 corresponds 12;
or by other correspondences 81, or -^/H, and so on. Now it is true that in
each case here mentioned we have a kind of correspondence, but these kinds of
correspondence are different, and herein lies the fact that all corresj.ondcnces
are processes. Equally, if we say that we have cases of relations,— that
3, 4, 7 stand in one relationship; 3, 4, 12 in another, etc.— these relations are
different, and the generic term for all of them is process. The psychological
fact that we may associate ideas together, and call such association, corres-
pondence, or relationship, functionality, or like terms, should not obscure the
mathematical fact, which is equally psychological, that we may pass from a
set of ideas to a different idea, or set of ideas, —a mental phenomenon which we
may call inference, deduction, implication, etc. We therefore shall consider
that any definite rule or method of starting from a set of ideas and arriving
at another idea or set of ideas is a mathematical process, if any person
acquainted with the ideas entering the process and who clearly understands
the process, would arrive at the same goal.
Thus, all persons would say that 3 added to 4 gives 7, 3 multiplied by 4
gives 12, etc., wherein the words add, multiply, etc., indicate definite processes.
330. Definition. A mathematical process is defined thus :
I. Let there be a class of entities \a\.
II. Let there be chosen from this class n— 1 entities, in order a^,a. a„_^.
in. Let these entities in this order define a method, F, of selecting "an
entity, a„, from the set.
Then F{a^, a, . . . . a„_^, a„) is said to represent a mathematical process.
' B. RlSSELL 1, p. 106.
76 SYNOPSIS OF LINEAK ASSOCIATIVE AliGBBRA
The entities Oj . . . . «n-i are called the first, second (n — l)-th
facients of the process. The entity a„ is called the result. Occasionally this
process has been called multiplication, aj . . . . a„_ibeing called factors.
331. The class of entities \a\ may be finite or transfinite. If transfinite
they may be capable of order, and may be ordered, or they may be chaotic.
It is not known whether there is any class incapable of being ordered, or not.
The number n may be any number, finite or transfinite, of a Cantor
ordinal series of numbers.
332. Definition. Let us suppose, in the process F {a^, a.;,--- -cin-i, <^n)) that
a„ is known, but a^ [l < r< Ji — l] is not known. We may conceive that by
some process F^, we can find a,, the order of the known terms being, let us
say,
K, «i., (^i^u «r)
where ii, to- ■ ■ ••i„-i are the subscripts 1, 2. . . .r — 1, r -\- \ . . . .n m some
order, so that
^.K, «i, «i.-,, «,)
F„ is called d>. correlative process, the c-correlative of i''. The process i^'is uni-
form when, for all correlative processes, cir is determined uniquely.
333. Theorem. There are for F, n! correlative processes, including F.
We may designate these by the substitutions of the symmetric group on n
things ; so that if we have
^(«l, «2, «3 ««)
then we also have
where a is the substitution
3..
334. Theorem. Evidently the cr^-correlative of the cTg-correlative o^F is the
cTj-correlative of -F, where
(Tg = a^ (T2
We write, therefore, F,-\ ,-1 = F„-\ = F^,^,„^-\ .
The correlatives thus form a group of order n ! .
335. Examples.
(1) Let as be tax-imijer, «i be hoy, a^ owner of a dog, then
F (oj a^a^ : a boy who owns a dog pays taxes.
F(\%) («i02«3) : the possession of a dog by the boy requires payment
of the tax.
F(iz) («i ^ f^z) '• the tax on a dog is paid by the boy.
■^(23) (^1 02 Og) '• the boy pays taxes on the dog he owns.
F^■^2;s) (^1 ^2 (^s) '• the tax paid by the boy is on a dog.
■^{132) (^i<^2'*3) '• the dog requires that a tax be paid by the boy.
(2) Let aj, ttj, 03 be numbers ; F{a^^aza^ mean a^ is the a^ power of Oj.
GENEUAL THEORY OP ALGEBnA 77
Then ^(,o) {uicuug) means a^ is the log r^^ power of the exponential of a,.
-^'(13) ("i^u";!) means a, is the quotient of loga^ by log a^.
i'^ia;), (ajaoCta) means a., is the a^ root ofag.
F,y>;n (ttj «., U;,) mcaus ttj is the log a, power of the exponential of - .
-^(133 ("i ";; '^'.i) 'neans Wj is log a^ on the Vjase a.^.
336. Theorem. The correlatives of F fall into sub-groups corresponding to
the sub-grou])s of the group (r„, .
337. Definition. It may happen that in a given process, F, we may have
simultaneously for all values of cfj. . . -w,,-!
^(«i, «2 ••••«») (1)
i^K,«i- ■••«») (2)
Since we must have from (l) i''^-! («(_, a,-,^. . . .a„) we must identify F and
F^-\, or as we may write it, F= F„-i . Tlie correlatives will break up then
into — groups where ?« i.s tlie order of the substitution a. We call these
cases limitation-types of F.
Examples. For i^(aj n.^ we have but one case : F=. F^^o,).
For F{a^ a^ a^ we have five types :
(1) F^=Fiyi). This is the familiar commutativity of ordinary algebra.
It follows that
-^^(13) ^^ -'^(23) -f^(123) ^ -^(132)
{2) Fz= J?'(,3„ whence i^j^, = F^^^-,, F^^^ = F^-^-,
(3) F=F^^^, whence i^,i,, = i^^^oj,, F^,^, — i^^ig.,,
(4) i?'= i^(i2.3, = ^(132), whence i^dj, = F^,^^ = i^^ag,
(5) i^= i^(i2) = i^(,3, = 7^123, = i^|i23) = i^(i32)
For i''(ai ttj c(3 aj we have twenty-nine types corresponding to the sub-
groups of the group G^, :
(1) F= F,,,, (2) F= F,,,, (3) F= F,,,,
(4) F= F,,,, (5) F= F,,,, (6) i^= F,^,
(7) i^=i^(j2)^34) (8) i?'= i^„3)(2j) (9) -^=i^u),23)
(10) i^=Fa23, =^(i=c) (11) F=F,,,,,= Fa,,, (12) J?'^ i?',,3., = i?'.„3,
(13) i^= i^(23,) = i^(243) (14) J' = i^(i234) = i^(]3) (24) = -^(14321
(15) i* = -rii324) = -r,]2| ^34) = -^,1423) (16) J' =■ /'(i:j42) = i'm) (23j = -^(1243)
(17) i^ = i^jjo) = i^;34, = i^(i2) (34) (18) F=z F^^3i = F^^i) = -^usj (24)
(19) i''= i''(H| = i^(23) = i^Ui (23) (20) F=Z i',12) (34) = i\i3) (24) = -f'iu) (23)
(21) F= i^(,2) = i^(i3, = -^(23) = -^(123) = -^(132)
(22) F= i^|,2, = i^(i4, = i^(24) — i^,i34) = i^(i42)
(23) F= i^(i3) = i^i4, = J^(34, = 7^,134, = i^i43)
(24) F= i^(23) = F^2l) = i^i3t) = -^(23i) = -^1243)
(25) J^ = F^^^| = F^o^^ = F^i^-^ (24) = -''(1234) ^^ -^(12) oo = -^(U32) ^^ -^lU) (23)
(26) 7^^:= i'^ij) = i^34) = -r\i2)(34) ^= -^(1324) ^^ -^(13) (24) = -^1423) ^ -^(14) (23)
(27) F^ F^^^^ = i'iga) = F^^^ ^3^■^ = -rjiigs) = -P(W) (gs) = -f^i824) = -^(13) an)
78 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
(28) i^= Fq2) (34) = -^(13) (24) ^ -^(23) (14) = -'^ISS) = -^'^(132) = -^(124) = -^(142)
-— -^(134) ^ -^(143) — - -''(2ai) ^ -''(243)
(29) i^=all
338. Theorem. It is evident that every group defines a limitation type for
an operation F of some degree.
339. Definitions. Suppose that in a process all the elements but two are
fixed, and that these two vary subject to the process. Then the ranges of
values of these two are said to form an involution of order one. If all but
three elements are fixed, the ranges of these three form an involution of order
two. Similar definitions may be given for involutions of higher order.
An involution of order r is often called an implicit function of r + 1
variables. The symbol consisting of the process symbol and the constant
elements is called an operator.
If in any involution of order one the two elements become identical so
that they have the same range, for any given set of constant elements, then
this set of constant elements constitutes a multiplex modulus for the process.
For example, in multiplication F.{ah = h) when a is 1. A similar defini-
tion holds for higher involutions.
If in any involution of order r, the constant terms determine an involu-
tion whose terms may be ani/ elements of the set, then the constant terms
constitute a zero for the process. For example, if F. (Oa = 0), for all a, is a
zero for multiplication. An infinity is, under this definition, also a zero.
We have seen that there co-exist with any process i^ certain other correla-
tive processes on the same elements. These give us a set of co-existences
called fundamental identities; but we may have co-existent processes which
are not correlatives. In the most general case let us suppose that we have
F' .Uiittiz «i„, F" .a^^a^ az^, -?""'"" -flr+i,! «,-i, a ^''•-i."r_i
and that when these processes exist, then we have F^'"' . a^.^ a,., a,.„, .
We say that i^^''' is the implication of the r — 1 processes preceding. We
enter here upon the study of logic proper. For example, if the processes are
F' . ah F" . ho F'" . ac
we have the ordinary syllogism.
We may symbolize this definition by the statement
* . i^,'„ F'l_ FtZ^ F^C
and we see then that the form is again that of a process *.
We can not enter on the discussion of these cases beyond the single type
we need, called the associative law.
Let F be such that for every a, h, c, we have
F.ahd F .dee F . beg then F . age
then F is called associative. The law is usually written ah .c = a. be
Processes subject to this law are the basis o( associative algebras.'
'Cf. SonaoEDEB 1; Rcssbll 1, 2; Hathaway 1.
=
PART II. PARTICULAR ALGEBRAS.
XrV. COMPLEX NUMBERS.
340. Definitions. The algebra of ordinary complex nuinbers possesses two
qualitative units, eQ= 1, ami e,, such that
The field of coordinates is the field of positive and negative numbers. The
field naturally admits of addition of the units or marks.
341. Theorem. Tlie characteristic equation of the algebra, as well as the
general equation, is
'^^ — 2x^ + x" + if—0
or
— y icco— ^
Hence for any two numbers
^(T + (T^ — 2X(T — 2/^ + 2xx' -f 1>JlJ =
or
^<T — xa — xX + xx' -\- yij -=.0
The characteristic equation is irreducible in the field of coordinates but
in the algebra may be written
(^ — a-Co — ye^) (^ — xCq + ye^) =
The numbers ^ =: arco + ye^ and iT^ = ^= a-Co — ye, are called conjugates.
Hence ^2 _ 2x ^ + x^ + / = has the two solutions ^, K^, or (^ — ^)
(^ — KQ are its factors.
342. Theorem. If several algebras of this kind are added (in the sense
defined by Scheffers) we arrive at a Weierstrass commutative algebra.
343. Theorem. If the coordinates are arithmetical numbers we must
write this algebra as a cyclic algebra of four qualitative units
Cq Ci e., eg
where
e\ = ^2 e? = 63 e\ = e^
In this case the units Cq and e, are not independent in the field, and com-
bine, by addition, to give zero,^ and the algebra is of two dimensions.
' Study S, and references there given; Beuax 3; Bellavitis 1-16 ;■ Bibliography of Quaternions.
79
80 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
XV. QUATERNIONS.
344. Definition. Quaternions is an algebra whose coordinate field is the
field of positive and negative numbers, and whose multiplication table is
(«o=l)
^0
ei
«%
^3
Co
'-0
ei
^2
%
^1
ei
—Co
^3
— ^2
^2
^2
—es
— ^0
ei
eg
«3
ez
— ei
— <'o
345. Theorem. The characteristic equation is
^ — 2xo^ + a-o + XI + icl + a| =
The characteristic function of ^ is
f— 2xo^ + ^^ + :^ + aH-a:i
If we define the conjugate of ^ by
then the characteristic function factors into (^ — ^) (^ — Z"^).
346. Theorem. The first derived characteristic function of ^ and a is
(. -Q{a- K^') + (cT- ^0 (^ - ^0
This vanishes for ^ v y/
'^'' ^ = /f^ a = K^'
347. Definition. The scalar of ^ is ^^ = .to = ^ (^ + ^). The tensor of ^ is
given by (7'0^ = Cr= ??•
348. Theorem. We have
^T + T^ — 2^S^'— 2rS'( + 2FF^7^' + 2 FFtF^ + 2 ^^If' =
if ^ = ^, and T = ^'; or if ^ = ^, T = ^ ; also S^^' = S^^'.
r
349. Definitions. The versor of ^ is U^ = -^ The vector of ^ is
350. Theorems. ^ = >S^ + F^^ ^ — S^ — F^
( T'O'- = ( ?t? = (-^O^ - ( VO' = {S^f + { TV'(f
S. '(0 = 801: S. K^ = KS.( = S^
If S(=0 = Sa a = Va ^ = V(
and F^. Fa = — Fct7^+ 2^.FffF^
QUATERNIONS
81
Also KV^ = - F^ = VJit;
'( . Va^-' = — Va + 2jS(;.Va.^-'+2 ^-' JS^Va
z= — Va + 2S(.V^^ + 2V.^-' ,Sr(V>y
If a, /?, y are vectors,
F . a V(3y = ySafS — pSay V . a^y = aSi^y — (iSya + ySa^
If ^ is a vector,
hSaPy = aSpyh + pSyah + ySa^h = Va(3Syh + VPySah + VyaS^h
V Fa/? VyS = hSa^y — ySa(3b = aS/Syb - ^S<x.yh
If a, h, c, d, e are quaternions, let us use the notation
Also
Then
B .ah = i {ab — ha) B<. ah = bSa — aSb
B .ahc = S . aBhc — V{aBhc + hBea + cBah)
B . ab = B . ah = V . VaVh = — B . ha B .hh —
S . aBbc = S . VaVhVc= — ShBac = etc.
B . ahc = — B . hac = etc.
S . aBbcd = — S . hBacd = etc.
e;iS' . aBbcd =. aS . eBbcd — hS . eBcda + C/S . eBdab — dS . eBahc
=^ — Sde . Babe + Sae . Bbcd — She . Bcda -\- See . Bdah
c d e
Sac Sad Sae
She Sbd Sbe
B . ahBcde =
B {Babe Bdef Bghi) =
B'{BabcBdef) = BefSaBbcd + BfdSaBbce + Bde SaBhcf
B {BabcBdef) =— BefSaBbcd — B'fdSaBhce — Bde SaBhcf
ahc
SaBdef SbBdtf ScBdcf
SaBgJd ShBgJd ScBgJti
I Saa', Sbh\ Sec' I ^ — SBahc Ba'b'd
I Saa', ShV I = SB'ah B'a'h'— SBab Ba'U
I /Saa', SbU, Sec', Sdd'\ = — SaBhcd Sa' Bb'c'd'
The solution of the equation aip +^a2 = c is
pz=z.{a\-\- 2aj Sa^^ + a„ Ui)~^ (cTj c + ca.^
351. Definition. If the coordinate field contain the imaginary V — 1, we
may have for certain quaternions the equation (f = 0, whence q = yd, ()~=. 0,
and y is any scalar. In this case there is an infinity of solutions of the equa-
tion in the algebra.
82
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
Also if (f — 20^0 g- + jcq = 0, then q-=.Xf^-\- yQ.
The nilpotent Q is always of the form a + */ — 1 /?, where
a^ = /^- S.aP =
Since V — 1 will not combine with positive or negative numbers by
addition, we may say that this algebra is in reality the product^ of real
quaternions and the algebra of complex numbers, giving the multiplication
table
€o
ei
^2
e^
^4 ej
«6
(°7
eo
^0
ei
^3
es
^4 ^5
^6
Cj
«!
ei
— eo
«3
— ^2
es — e*
^7
— ^6
«2
e..
— ^3
— eo
f"!
es — e^
— e*
^5
ez
^3
^2
— ei
— eo
^7 +^6
— ^5
— 64
«4
ei
fs
^6
^7
Cq ^I
^^3
— ^3
^5
^5
— ^4
^7
— f-e
— ei +^0
— es
+e.
<?6
•"e
— ^7
— e*
<'5
— ^2 +^3
+ ^0
— ei
e-i
er.
+ ^6
— ^5
— e^
^3 — ^2
+ ei
+ ^0
with equations of condition
eu + e* = cj -f cs = e^ + eg =
This algebra Hamilton called Biquaternions.
^3 + f 7 =
352. Definition. The algebra ?.jio, Xjoq, /Lojo, ^220 fi^so is a form to which
real quaternions may be reduced by an imaginary transformation.^ By a
rational transformation this becomes
'^uo + ^'.
220
?.
110'
^220
^^210 T '^12(
120
Aoin ~~" Ai
120
References to the literature of quaternions would be too numerous to give
in full. They may be found in the Bibliography of Quaternions. In particu-
lar may be mentioned the works of Hamilton, Tait, and Joly.
I That is, any unit may be represented by a double symbol of two independent entitles, the two sets
of symbols combining independently.
' B. Feirce 3.
ALTERNATE} ALGEBRAS 83
XVI. ALTERNATE ALGEBRAS.
1. ALGEBRAS OF DEGREE TWO, WITH NO MODULUS.
353. Definition. An ulternate algebra is one in which the defining units
are subject to tlie law
The product e^ Cj = e\ is variously defined. In the simplest cases c? is taken
equal to zero.
354. Theorem. When Ci e^ + e^ei = 0, i, y = 1 . . . . r, we have
e? = 1=1..
r
^2=0, ^(T + <T^ = all values of ^, a
^ a T = all values of ^, a, t
For
and'
^(T . T = ^ . Tff = TT . ifff = — T . ^(T =
355. Theorem, We may therefore select a certain set of r — m — h units,
ej . . . . Cr-iH A> whose products CiCj^i, j= 1 . . . -r — m — h) are such that at
least one for each subscript does not vanish ; we may then choose for the next
m units the ?n independent non-vanishing products of the first r — m — h
units; finally, the last h units may be any numbers independent of each other
and the first r — h units. W^e must have ^
„, < 2(r-h)+l-VS{r- h)+l
2
or
(r — m — hy — (r -\- m — A) =
2. GRASSMANN'S SYSTEM.
356. Definition. The next type of alternate numbers is that of Grassmann's
Aiisdt'linungslehrc. In this case there are m units wbich may be called funda-
mental generators of the algebra, Cj . . . . e,„. For them, but not necessarily for
their products, the law e^Cj -\- ej 6^-:= {i, j ■=■ I. . . -m) holds. They are
associative, and consequently the product of ttj + 1 numbers vanishes. There
are r = 2'" — 1 products or units, e,, e^e^, CjCje^, etc.
This algebra uses certain bilinear expressions called products, which do
not follow the associative law, and also certain regressive products, which do
not follow this law, and which are multilinear expressions in the coordinates
of the factors.^
1 ScHEFFERS 3. Cf. Caucht 1, 3, S ; Scott 1, 2, 3. <ScHErFBBS 3.
•References are too numerons to be given here. In particular see Sibliography of Qualemiont;
Grassmann's works; Schleoel's papers; IItdh 1, 3, 3, 4, 5, 6, 7, 8, 9 ; Beman 1; Whitehead 1. Cf.
WiLSON-GiBBS 1 ; Jahnke 1. Works on Vector AnalysU are related to this subject and the next.
84 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
3. CLIFFORD ALGEBRAS.
357. Definition. A type of alternates of much use, and which enables all
the so-callecl products of the preceding class to be expressed easily, is that
which may be called Clifford's Algebras. Any such algebra is defined by m
generators e^. . . .e,„ with the defining equations
ef= — 1 Ci Cj -I- e, Ci = »,/=! m; i:^j
Ci . Bj e^ = Ci Cj . e^t i, y. A; = 1 . . . . m
The order Ms r = 2'".
358. Theorem. If m = 2ni', m' an integer, the Clifford algebra of order
2'" = 4"' is the product of m' quaternion algebras. It' m = 2m' -{- 1, m' an
integer, the Clifford algebra of order 2™= q^"'"*"^ is the product of m' quater-
nion algebras and the algebra^
2 __ o ^^ __^
359. Definition. Since any product such as
^i, ^i; • • • • ^ii • • • • ^i, • • • •
may be reduced by successive transpositions to a product of order two lower
for every such pair as e^, . . • -e^j, it follows that in the product of n numbers
^1- • • -^nt where ^4=2 x^e,, we may write
S 1 SZ • • • • S n = T n • S 1 • • • • b II 4" •« - 3 • S I • • • • Sn + • - - •
< J (n + 1)
8 =
The expression F„_2j, is the sum of all expressions in the product ^i....^„
which reduce to terms of order n — 2s. Evidently when n is even, the lowest
sum is a scalar, Fqj when n is odd, the lowest sum^ is Fj.
360. Theorem. To reduce to a canonical (simplified) form any homo-
geneous function of A'' of the m units, consisting of terms which are each the
product of a scalar into n of the units (of order n, therefore), we proceed thus :
Let q be the given function. Then by transposing the units, we may
reduce q to the form
q= — q' ii + q"
where i\ is any given unit, and q' (of order n — 1) is independent of q" (of
order n) and of ij. We find easily
q' = F„ _ J qii q" ij = F„ + i qi^
Therefore
Fi . <?2' = Fj <? F„ _i qii = wj = O (ii)
The linear vector function <!> is self-transverse, has therefore real, mutually
orthogonal axes. These are the units to be employed to reduce to the canon-
'For this class see Sihliography of Qualernions; in particular Clifford's works; Beez 1 ; Lipsohitz 1;
Jolt 6, 12, 25 ; Cati-et 6, 7.
'TabbrI. 'Jolt 6.
ALTERNATE ALGEBRAS g5
ical form. For example, if q is of order 2, and the function is the general
quadratic for N units, there are i N{N — 1) binary products. Then
— m^= F] qi Vq^i = — ^i ™i + V^q' a =: — V^q Vj qi^ = — 4>ti
If ij is an axis of this equation so is Wj. Hence the quadratic takes the form
q =z aj3 ^1 *2 + «(n ^3 *4 4" • • • • + (tzp - 1, 2p hp - 1 hp
where 2> = ^ N or i {N — 1) as iV^is even or odd.'
361. Definition. Let K change the sign of every unit and reverse every
product. Tiiea ii' q is liomogoneous, of order p,
K.q, = {-yr><v + ^^q,
Hence K. qp= ± q^ as /) = or 3 (mod 4) or = 1 or 2 (mod 4).
Let /reverse the order of products, but not change signs, thus
I.gp = qp if p = 0, or 1 (mod 4)
/. qpZ= — q^^ if J) = 2, or 3 (mod 4)
Let J chang'e the signs of units but not reverse terms.
362. Theorem. K. pq = Kq . Kp I. pq = Iq.Ip I =JK= KJ
J = Kl = IK K = 1J= J I P = J' = K-=1
363. Theorem. Let j> be of order 2, q of order 3 ; then
Pz ?3 = 1^1 • 2h ?3 + Fj . p2 ^3+^5. pz qs
Hence, taking conjugates,
— QaPz = — Vi- 2h ?3 + ^3 • p2 qz—V^. Pi 53
and
^1 • i'g S'a + "^6 • i'3 g-s = ^ {pi ?3 + qzP^ ^a • ih qa = \{ ih qs — qa Pi)
This process may be applied to any case.
364. Theorem. Let
q = q' + q" K.q = q'—q"
q.Kq=q'"'-q"^~-{^q"-q"<^)
Kq.q^f^-i'-'^islci'-ci'cl)
Hence
q . Kq= Kq .q if 5' q" — q" q' =
Let the parts of q be (according as their order =0, 1, 2, 3 mod 4)
q = ?(0) + ?(!) + qm + ?(3)
Then
q" q" = q^o) qa) + q^) ?(2) + q^o^ qi?) + q,z) qa)
and the condition above reduces to
9(") qm 9(1) 9(0) = q(2) 9(3) 9(3) 9(2) 9(0) 9(2) 9(3) 9(0) = 9(i) 9(3» — 9(3, q^)
or
^(3) (9(0) q^i) — 9(3) 9^3)) = 1^(0) (9(0) 9(2) — 9(1) 93)) =
'Jolt 6. This reference applies to the following sections.
86 SYNOPSIS OF LINBAK ASSOCIATIVE ALGEBRA
When this is satisfied
qKq = F,o, (9^0) — 5a) — ?(1) + ?J,) + 2 1^,3) iq,o) 9(3) — 9a) 9,3 )
This is a scalar if T^Q) — ^o ^^^^ F^^^ = 0.
365. Theorem, q- Iq = Iq -q if
^(0) (9(0) qrz) — 9(3) qa^ =0 = F^i> (g^„) (^^3, — g-,., ^d,)
366. Theorem. Let P= qpq~^, where q is any number, possibly non-
homogeneous. Then P=.V^i^ . P if qKq and qlq are scalars.
But F(i) may not =Fi. For example, let
q z=: cos ?< . t'l ?2 + sin ?{ . ig 1*4 ig ig
q~^ = — cos u . ii u + sin u . i^ i^ ij if,
q" =: — cos" u + sin- zt + 2 sin u cos u I'l i^ ^3 i^ 1*5 ig
p = ij, i., is, ii, if,, is, then
P = q\ q~u — 2^*3 — 5'\ — 5-% — <i\
which are of the form F^d z= Fj + Fg.
367. Theorem. An operator 5 O^"' can be found which will convert the
orthogonal set i^, i^. . . ■ i„ into any other orthogonal set ji,jo • • ■ ■ jn\ namely,
where
s, t, u . . . . = 1, 2 . . . . n s:^t:^u:^ . . . .
If qpq~^ ^=^Ppi>~^ foi' all values of p, a vector, then j is a scalar multiple
of p. q may be written
9l2 9»» 931-1.21
where 2l = n or 7i — 1 as m is even or odd, and
grj2 ^ cos ^ Mjo + h h sin ^ ?<i2, etc.
BIQUATERNIONS OR OCTONIONS 87
XVn. BIQUATERNIONS OR OCTONIONS.
368. Definition. Besides Hamilton's biquaternions, two algebras Lave
received this name. One is the product of real quateriiiotis and the algebra
6i : e5 = eo = e^, pq Cj = ejeo = Cj; the other is the product of real quater-
nions and the algebra ' 63 : e^= Cg, eg Sj = e, «„ = ''1, ej = 0.
369. Definitions. Let fl" = ; let q, r be real quaternions ; 11 is commu-
tative with all numbers; q = G>-{.x; r = a+i/. Then the octonion Q is
given by
Q = q + nr
We call q the axial of Q, Clr the converter of Q. The axis-direction of Q is
UVq. The perpendicular of Q is m = F . ctcj-^ The rotor of Q '\s Vq; the
lator is Fr; the »»o/or, Vq + H Fr. The ordinary scalar is iS'*/; the scalar-con-
verter is fl/Sr; the convert is *S'r.
We write
J/i Q = Vq M,Q = £iVr MQ = M, Q + M„ Q
m. Q = Vr S,Q = >Sq S, Q = flAV SQ = S, Q -\- S.. Q
s. Q = Sr M.Q = M,Q + £imQ S . Q = Sy Q -^ D.Iq
Let y, r, Q be the conjugates of q, r, Q, also designated by Kq^ Kr, KQ.
We define
KQ = Kq-\- D.Kr, or Q = q-\-D.?
The tensors of q and r are y'g-, Tr ; the versors, Uq, TJr :
The avgmenter of $ is 7g= r*? (1 -f nAS'r5-')= 7', Q.T.,Q=T,Q {l+HtQ).
The ^e«6o?- of g is T; §.
The additor of § is 7; g = 1 + nxS'ry-i.
The pitch of Q \s tQ — S . rq-\ T„Q=1 -\- flfQ.
The twister of Q is UQ = Uq {1 -^ D. Vrq-^} = tr, g . UQ.
The verso/- of Q is ZJj g = Uq.
The translator of $ is t^o § = 1 + 11 Fry-'.
Hence
g=7;g.7',g. f/-, g. tr,g
370. Theorem. Octonions may be combined under all the laws of quater-
nions, regard being given to the character of il.
371. Theorem. If Q, R be given octonions
Q-^R = X QE=Y
and if s is any lator; then if
Q'= Q -h £lMeMQ R = R+ (^ME}fR
then
Q' + R'z= X' Q< R' = Y'
'Clifford 1, -J ; M'Aulat 2, wbich applies to sections foUowing; Combebiac 1, 2; Stcdt 4 5
88 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
the application of ^, to all octonions gives an isomorphism of the group of all
octoaious with itself.
372. Theorem. If Q = ^, . Q', or q + D.r=^^ (q' + Hr'), then
q' = q r' = r — i/(p Mq)
373. Definition. The axial
q(i = q + D. {Mqy^ MMqMr = a: + o + 0(0"^ Mi^a
is called the special axial of Q, and
r^ = Sr+{\ + CiMMr ( Mq )-') Mq S Mr ( Mq)-^
- y + Tl + £iMM ^ ") ioS —
is called the special convertor-axial of §.
374. Theorem. We have
Q = q + nr = qg + Hz-g
"""^'^'^ q,= \l+M.{Mr[Mqr'){)\q =^M...^^q
r^ =r -; 1 + 1/ . (i//- [Mq]-') ()\lr—3£. {Mr [Mq]-') Mq)
= <pM .0.-1 (r + J/ . cjJfocj-^) = <?).v ,.-1 (aS/- — a-' *S'a)(7)
That is
$ = ^M. ..-1 ?', where Q' = q + D. {Sr — cj-' &>a)
or
Q'= q+n [Sr — {Mq)-' SMq i/r]
375. Theorem. Any octonion mny be considered to he the quotient of two
motors. That is, if Q be an octonion it may be written Q = BA"' or QA = B,
where A and B are a pair of motors.
376. Theorem. Q' = q-'—^q-'r q^\ when q:^0.
377. Definition. The angle of g- is fJie angle of Q.
378. Theorem. Q () Q~' produces from the operand a new operand which
has been produced from the first by rotating it as a rigid body about the axis
of Q through twice the angle of Q, and translated through twice the transla-
tion of Q.
379. Theorem. If A and B are motors
A = a^ + na2 = (l + Clp) a,
B=(3, + a(3, = {\+ ap')[i, = ] 1 4 n(m +/)}/?
and
then
AB = ai^ + a{p+p' — w)ai(S
M . AB = Ma,p + il \{p + p') Ma,p — w Sa,^\
M, AB = Ma,lJ m.AB = {p + p') Ma, l3—wSa,/3
tM . AB =p-{-p' — m M-' UiiSSui (3 =pi-p' + dcot 6
Hence axis M. AB is w, pitch =p \- p' -^ d cot d
BIQUATERNIONS OR 0CT0NI0N3 89
If A and B are parallel, we deteniiiiie M . Ali by
M.AB = — D.ma^^
Again S.AB = S.a,(i + £l\{ p + p') .S'a, /? — -^ Ma, [i \
S, AB = S<x, (3
s.AB=(j> + //) ,Va, (i—w Mai /?
tS.AB=p + j,'—m Mai t^ 'S~^ "1 f^ = l' + p' — d tan
Mi.AB + S,.AB = ai(i
m AB + sAB={pJr p' — m) a, P
tAB = p + p' u.AB = — m
T,.AB= T{u, (3) U,AB= U{a, /?)
For the sum we have
A-\.B = \l+D. (//' + u,')\ (a, + li)
where p" + w' = (p a, j- p' (3 + w 13) (a, + (3)-'
or
p"=S{ l> a, +p'l3) (a, + /3) -' - u, i/ai 13 . (a^ + /3)-2
m' = w Si3 (aj + f3r' + {p~ p') Ma, /3 . («! + /i)--
380. Theorem. If .4, B, C be three motors, and if cZ and are defined as
in §.379 for A, B, and likewise e, ^ are corresponding quantities for M . AB
and G, then
^ SABC= (A + tB + tC+d cot — e tan 4)
Hence if we have three motors 1, 2, 3, and if the distances and angles
are: for 23 :c/j, gj; for 31 : (7o, 0^; for 12 : cZg, 0;;, and fjr 1 and (Zi : e^ , ^,; 2 and
rfo : e.,, ^.r, 3 and d-^'.e^, <p3, tlien
d, cot 6i — Ci tan ^j = d.^ cot 0, — e.^ tan ^3 = tZj cot 63 — e^ tan ^3
381. Theorem.
T,{QR ....)= T,Q.T,R.... ( (QR ....) = tQ + tU +... .
S\Q-M\Q=T\Q tSQ.S\Q~tMQ.M\Q = tQ. T\Q
A .nrr, tQT\Q — tSQ.S\Q
and tMQ = -^gZT^Ig-^^
382. Theorem.
i Tir A T>ri J < I . r. I , ^ <^ cot Q — e tan A
cot-0tan-^ +cot-0 + tan-(^
tM. [MAB) C= fA + /5 + tV + (Z cot + e cot (^
383. Theorem. If £■ is coaxial with A, B, C, then
ES . ABC =AS.BCE-\- BS. CAE + CS . ABE
= MBC. SAE+ MCA . SBE + MAB . SCE
90 SYNOPSIS OF LINEAJl ASSOCIATIVE AT^GEBRA
384. Theorems.
S,{Q + E) = S,Q+S,R s{Q + R) = .sQ + sR
SQ = sQ.Q + £lsQ
If iljl k, then
A = — iSiA — jSJA — kSkA
or
A = — isD.iA — JsD.jA — ks£ncA — £iisiA — 0.jsJA — D.kskA
If
A =: xi -\- yj -\- zk ■{■ lD.i + mD.j + 7iQ.k
and
cs . d , . d ,,9,„.9.„.9 ^,9
^=^9T+-^9^+^9^ + "^9^ + "'^9^ + ^^^97
then
{s.dAb) = — d{)
S' is independent of i,j, k.
If ^ is a lator, s . A- =^0.
If A is not a lator
s.A" = 2tA3IlA = — 2tAT]A s{TQY= 2tQ . T\Q
385. Definitions. The motors Ai, A.,....A,i are independent when no
relation exists of the form
a;i J-i + .... + a^ii -^u^ 0, [xi- . . .x^ scalars]
If independent, the motors XiA^-\- .... +a-„J.„^2x.4 form a complex of
order n, called the complex of A^. . . . A^. The complex of highest order is
the sixth, to which all motors belong.
Two motors A^, A.^, are recijyrocal if sA^ A^ = 0. The ?i motors A^ . . . . A„
are co-reciprocal if every pair is a reciprocal pair ; in such case A^ is reciprocal
to every motor of the complex A^- ■ ■ -A,,, and every motor of the complex
Ai- . . .Ar to every one of the complex ^r + i- • • --^u- The only self-reciprocal
motors are lators and rotors. Of six independent co-reciprocal motors none
is a lator or a rotor.
386. Theorem. If .4, B, Cave motors, S.ABC=0 if and only if
(1) Two independent motors of the complex A, B, G are lators, or
(2) XA + YB -\- ZC = 0, where X, Y, Z are scalar octonions
whose ordinary scalar parts are not
all zero.
387. For linear octonion functions and octonion differentiation reference
may be made to M'Aulay's text.*
'M'Aii.AY a.
TRIQUATERNIONS AND QUADRIQUATERNIONS 91
XVni. TRIQUATERNIONS AND QUADRIQUATERNIONS.
388. Definition. Triquaternions is an algebra whicli is llie product of
quaternions and the algebra'
389. Definition. If /■ = ?/• + p + u («•, + pj + ^ {n-., + ^■,) = q -\- i^qi -\- ^q.^,
where q, qi, q^ are ordinary quaternions, then we write and define
r ■=■ w + (cjJi'i + jupa) + {^w-> + p + cjpi) =■ G . r -{- L . r -\- P.r=zw+l-\-p
where
G . r = w = S .q
L . r = nio., + p + opi = UfSq., + Vq -\- oS(/i, called a linear element ;
P . r = uwi + np.^ = (dSqi -f- ^ Vq-i, called a plane.
Further, we write
L .r=:{^ w., + o/?) + (p -f opi — u^3)
where we determine /3 by the equation
{vM — p") /3 = wl p + tvo Fppi — p«Sppi
then we define
m = (^ ir., + (j3), called a j^oint
fZ = (p + wpi — oi3), called a line
L .r = m -\- d
We define further
L . r ^m — d, the conjugate oi L . r
390. Theorem.
G.IV =G.l'l LAV =—L.Vl P .IV — P. VI
G .Ip =0 L.lp = L.pl P .Ip =-P .pi
G .pp' = G .p'p L .pp' = —L .p'p P.pp'=.
391. Theorem.
G . md ^ G . dm = P . md = P . dm L . md = — L . dm =
392. Theorem. Lr . Lr = m" — d^ l'^ = ..^ ,, T
m~ — d-
393. Definitions. T . r = \^ ic" ^ U - f
If
Yq. = P .r = c^Sqi = u? . r
or
P . r=Sqi, if Vq, =
394. Theorem. Let
A =■ w- + ll — p- =z q~j + q., q.,
B = 2{w Tm — TLpd) = qq., + qzq
then
r-i = {A- - B-) {{A — fiB) {q + /u7,) — 0) (^ - 2,) q,(q+ q.^)]
■COMBEBIAC 2. This reference applies to the foUowing sections.
92 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
395. Definition. Let
m ^ (ixq + ap 77?' = (ix'o + op' c = a + (j/3
Then we define
F . m, m' = o-Q p' — 3*0 p + oZpp'
/S . c, m = fi {a-Q^ -f- G^ap) + idGiSp
S . m, m', m"= jS{V . m, m') m" =i S . mV . m', m"
396. Theorem. V . wp, op' = oLpp' /S . op, op' =■ oGpp'
S . c, m=: S . m, c V . m, m' = — Vm', m
G . c S . c, m =■ L . c S .c, m =1 iPc^. tti P . c S . c, in = S . c, Lcm
G . S . c, m . m ■= L . S .c,m .VI :=V. in, Lcm P.S .c,m . m =
G . ni V. m, m'^ Z . 77i Vm, m'= P. mV. m, m'^ — *S'. m,Lm m'
397. Theorem.
GW = a-o a;^ + %'
Lir = 7pp' + o [ V{pp{ + p, p') + Xq Pi — xo pi]
PW = II {x, p' + x^ p) + o>S'(ppl + pi p')
i77l 77l' = a^o Pi ^0 Pi
398. Theorem. Gpp' =— Tp Tp' cos {p,p') Lpp'= Tp Tp' h sin {p,p')
399. Theorem. I =■ ^Xq 4- p + opi p = «a + oi« Glp =
Lip =: jU/Spa + a:o a + o (T^jp -\- Vpi a)
Let ^, o, o' be units satisfying the multiplication taVjle
^
o
o'
o'
1
o
— o'
— o
-2(^+1)
1
(J
2(^-1)
and let the quadriquaternion^ A be defined bj the equation
A = q + (xqi + aqo + Jq^
where q, q^, q.i, q^ are real quaternions. The units i.i, o, o' are commutative
with q, ji, q.,, q^. If ^'3 = 0, A becomes a triquaternion.
We may write A as the sum of three parts each of which may be found
uniquely :
A=G.A + L.A+P.A
where
G.A = S.q
L . A= F. (7 + //iS'. <7, + o F. q. + o' F. (73
P .A = (U F . (/i + oA\ 5-0 + o'*S' . 5:5
Then the formulas of §.390 above hold for quadriquaternions as well as for
triquaternions, if ^ ^ Z . A, p = P . A, etc.
' COMHKlilAC 3.
SYLVESTER ALGEBRAS 93
XIX. SYLVESTER ALGEBRAS.
1. NONIONS.
400. Definition. Nonions is the quadrate algebra of order 9, corresponding
to quaternion-s, which is of order 4. In one form it.s units are*
''-no ^\:lO ^IM \'10 ^-^M ^iM ^:m ^^820 ^^330
401. Tlieorem. The nonion units may be taken in tlie forms (irrational
transformation in terms of co, a primitive cube root of unity)
^0 — - 1 — - '^iin T ^^220 + ^x]o ^ ^^ ^no T <^^220 i" '^'''''•aio * — ^ '^iio 4" 'J"''>-220 "1"^ ''■330
J •— '^l-.'O ~r A^.30 + XjiQ ^" = Aj3o + Aoio + /I320 y ^^ ^^120 "i" ''-' '^230 "F'J'^aio
V -^ /l];jO + oPlojq +(J"A3:io fj = AjoQ +0-/l;>3() + (')\-i|0 ^7" ^— ^130 "H "'\'I0 ■(""''•320
whence"
t^^zi y3^i (y? = i (yT=i (5T= i (*Yf=i
402. Theorem. If
<? = 2 a-„6 iV" «, 6 = 0, 1, 2
Then
S .J <p = Xq2 S .if (p =: I^Xjo S . i'j ^ := (J X12
S .y^ = a-fli S . y "4> = "a"2i S . i^j'^ = orxn
S.^^ = S{Exa, iV) i^Uca i'j") = SX X,, y,, a."= i"+=y*+<^ (?+,'5S) (,nod 3)
= (a-oo Z/oo + a-10 Z/20 + ^-xi 2/10 + a^oi 2/02 + "'a;ji 2/22 + ua;,! yjg
+ a-02 2/oi + "^12^/21 + "°a;o2?/i,)
Hence
and if
therefore
^■l = 2x„j7/,,G)'^i« + ''y* + '' a, J, c, fZ= 0, 1, 2
403. Definition. US.j^=0 S.j-^ = S.j=0 then we
define
404. Theorem. We may write ^ in the form ^ = a + />*' + ci" (at least if
^ lias not equal roots); whence, ify is chosen/ so that Ay=0, S.ji^O,
Sj-i = 0, we have
A'<^ :=Jcpj-^ z= a + wi/ + cj" cr -S'|<|) =J'^J~" = (I + (■i'i'i + "Ci"
' Sylvester 3, 4 ; Tabek 2 ; C. 3. Peirce 6 ; also the linear vector operator in space of tliree dimensions,
Bibliographij of Quaternions, in particular Hamilton, Tait, Jolt, Shaw 2; also articles on matrices.
» Shaw 7. This applies to §§403-403. » Of. Tabek 2.
94
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
405. Theorem. If y is to be such that S.j' = 0, Sij' = 0, Si"j' = 0, and
if y is such that S .j = 0, Sij =^ 0, Sij^ = 0, we may take
J' = O.J + a J- + /3i ij + /?3 if + Y\ i'V + Y-Jt
whence
y- = 2ai tto + 2/?i 73 u- + 2/3. yi CO + I (/?i aj + o' |8i ag + a^ /?3 w + aj /Jg)
+ i" («i ^2 + «i 73 "' + «3 7i " + 0^2 yO +
and if S .j''' = 0, Sij'" = 0, ^'ry = then
tti tto + u iSs 7i + 0)" /3i 72 = ^i Uo + t'> «i f^n =■ tti 73 + (J Ua 7i :=
whence
ar =2/^1X1 a^=2/33 73
and
That is
Hence
and
"3 • 1^3 • ^^3 -— <^i • — (j" /^i : • — 071 or a^ = /ig = 72 =
J' = («! + /^i i + 7i i')j=j{(^i + "' /3i i + 071 i')
j,-i _y3 ^^^ + /3j i + yi i2)-i = (aj + CO- j3i i 4 (071 t-)y-i
y (a + &i + ci")j'-^ =j{a + bi + ci^)j-'^
It is thus immaterial what vector/ we take to produce the conjugate Kj^,
except that we cannot discriminate between Kj^ for one vector and Kj^ for
another, if the second is equivalent to the square of the first. We may
therefore omit the subscript y and write simply K, K".
or
406. Theorem. From ^ =■ a + bi -\- ci^ we have
<^3 _ 3a ,p" 4- 3 ((t^ — be) ^ — {a^ + b"^ + 0^ — 3 abc)
S . ^ — ^ S . i^ S . i^(p
S . i~^ S . <p — ^ S . i(p =
S . i<^ S . i'<p S . <p — <p
407. Theorem.^
^f\<p + ^lC~<p + I\(pK''(p = 3 {S"<p — Si^ *S'i-» = 7;<^
^K^K-^ = S'^ + SHcf, + SH'^ — 3 Sp Si'^ Sicp = T,^
T,^= 7\ A> = 2\ K'<p 7; 4> = To A> = 7; A'"> 7*3 <?) = T's 7v> = T, IC~^
408. Theorem. If a — 1 + i + i", where «^ = Sa-^b «"/",
S . a^ — <^ S . j~^ a(p 'S' . y ~- a<^
.S' . aj<p S . y-i aj<p — 4) /S" . y-2 aj^ =
yS . aj'p /S' . y-' ay-^ — ^ S . J-- aj~^ — cp
iCf. Tauek 2.
SYLVESTER ALGEBRAS
95
7; ^ = >S'(a + ,/-' aj +y-2 af) ^ = S (a + ICa + IC'a) <p
— Aay*?* 'V'^'^^P — Aay"</> >Sj ''a<lj — >SJ'<xj"'^ 'V~'''^Jl')
409. Theorem.' <p^ — SS^ . ^' + | {Z>S~^ — t^^') ^ — (| /S'>
</'i ^li + <?»i ^:; ^'i + ^i ¥i — ^'^i • (4*1 ^■'. + '/'a <?'i) — 3'^'<?':: • ^1
+ 3 {S-<pi — i SV^^^^)^., + 3 ('2A'(^, A'(^, — A'F<^jF^.,) <^,
— (3/S'-<^i S(p2 + 3A' F-'(/;, F^o — Z>S<pi >SV<p^ V^.. — I *S'(^. 'S'F=^ij =
where V^ = {^— S^)
Also
4>1 'pi 'P3 + ^l <?>;) ^i + <?>:; <?'l 4>3 + <p2 4»3 <?>1 + 4*3 ^I ^'2 + <?'3 'Pi ^l
— SS^i . {(p2^:t + 'ps'Pi) — ^S^j ■ («?>i4>:) + ^.ifpi) — 3'S'(^;! . {^I<p2 + ^2'Pl)
+ 3A'<^i (SS(p, A<^3 — &'^, ^,) + li^, (3A'<?)i Sep, — .Scp^ ^,)
+ 3(j);l:iS<pi S^., — S<pi ^.,) — {27S<pi S<p.. iS(p:i — 'JS<pi >S^.. ^;j
— QS<p., S^i ^3 — SiSi^a S(pi ^0 + '6S(pi <^o <^3 + 3 A"!:^! (^3 1^,) ^^
410. Theorem. If
/; {6, y;) = h [e" + " + 0)^6"' + -■-■" + 0)=''' e^^^ + ""J /.•= 0, 1 , 2
then
^ = {n ^y [/o (^^ ^) + i A (0, >7) + i' h {6, 0]
= (7'3<?>)* [Me, 0) H- i-/, (0, 0) + rf, {0, 0)] [/o (^, 0) + (A (>:, 0) + r/. (>:, 0)]
If ^1 and ^i have the same unit i,
^1 = a + bi -\- ci- (^o = a' + 6't + di^
^i^2 = {Ts^iy{n^-^'[fo{e + e',y! + y!')+if,{0 + 6\-r + y;') + i;%{d + e',-^ + r:)-]
The functions/., satisfy the addition formulae '
A.(e + e',y: + y:')=Me, r)MQ', rj) +/,(9, n)f,^,{&, rj) + /:.(e, >:)/,+, (6', rj)
/, (u0, 0) = «y, (0, 0) A ("0, co^V) = coy, (0, >;) A (e, 0) = y;, (o, 0)
K" . ^= ( n^y [/„ (6)'^(j, <o,7) + (/; {co% 0)^) + t-y, (o-e, u^)]
<?,-' = (73<?>r* [/o(- e, - >:) + if, i-d, —r) + i% {-6, -r.)]
r = in^r'l /o (po, py;) + */i {pO, p^,) + ^y (;^e, !'>:)]
411. Theorem. Tlie characteristic equation of <^ = S a:„6 i" j^ may be
written
^00 4- 3-10 + a-,., — (?)
Xyo + w a',2 + o'-Vjo
+ a-i, + a-.j
•^'o; "t" •'■jo ~r CToo
Xuo + (J a-10 + u"a:"2o — ^ -i'oi + "3-11 + u'-'x,!
a"o3 "I" ""•'•j2 + CO avj a^oo "t" <^'a'iQ -)- w a-oQ — ^
=
■Taber 2.
96
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
The general equation is tlie cube of this, but may also be written
ABC
where
CAB
B C A
=
A=
^00 'P •'^10
X.
20
20
X
10
Xqq ^ XiQ
^20 ^00 *?• ■
21
B= \
6) J-ji CJ J'oi O Xn
o o o
c=-
2^02
a-v
ti) •^22 ^ "^OS '^ *^13
412. Theorem. The cubic in ^ has three roots, corresponding to which, in
general, there are nine numbers, in three sets of three each, such that each
set is multiplied by a root of the cubic when multiplied by <^ ; if these are
pn. Pi2, Pis; p2i, p22, P23; P31, p32. p33, then Qk being the root corresponding to
the A;-th set,
4> • 9ki = Ok Pki
413. Definition. The transverse of cp-='Z;r^^i'^p^ as to the ground defined
[f ^ =. ^j (p is self-transverse.
414. Theorem. We have ^<p = (p(p, 4*^^^^, so that ^^ and ^(p are self-
transverse. Again <p4^ =^ 4'^.
For
hence
Also
thence
^4>=^ x„b Z/,_„,„-b (o"^-^-'" i'j-"
^ = a-„, (.-"'*" y-*
^^ = 2 a:„, y,, uT'^^^'^-'^'i i«+=y-*-<*
When
415. Theorem. We have
<?)o = M<?> + ^) = i 2 {x,, + (."^ x,,. _,) i"y*
i (<?> — ^) = i S (a:„„ - o"^ a;„, _,) i^y^
ab ,
<?> = <?> a:<jb= w'"'x„._i,
1 For further theorems nnd applications sec Joi.Y 1, 2, 3.
SYLVESTER ALGEBRAS 97
2. SEDENrONS.
416. Definition. Sedenions is the quadrate of order 4'. Its units may be
expressed by
'^•ijo i,jz=\ .... 4
It may ahso be expressed by units of the form '
V" /'' i'' i'''
where a, b, c, d = 0, 1, and
^1 *i "^ ~ ^1 ^1 iz h = — h h h is = iz h h h = h h
ji h = hj'i j\ h = h h i\ = j\ = il =jl = — l
417. Theorem. Sedenions may also be expressed in the form
J = ^m + \':!o +- Xaio + '^iio e^,, = Cj
a,b=z0,l,2,3 ji=zs/'^ij
418. Theorem. If
419. Definition. If ^S'.y=Oand
S.j^ = = S.f<p = S.f^ y*=l
then we define ''
420. Theorem. We may write generally (that is, when ^ does not have
equal roots, and in some cases when it has equal roots)
no
— ,•« ^'h
I
Whence
Accordingly
^ = a + hi -{- cp -{- dP
I\(j) = a + bd — ft- — dii
IP^ = a — bi + ci- — di
IPcp = a — hii — eP + dd
3
3
'3
bi =^{^-— iK^ — K-^ + tA' »
di^ = i (<?) + iK<p — K"-^ — iK^^)
421. Theorem. Theorems entirely analogous to those for nonions (see
§§402-415) may be written out.
422. Definition. The transverse of = ^x^^ i"/'^, as to the ground defined
^y *')/» i« defined to be
' Sylvester 3; Taber 3 ; C. S. Peirce 6 ; Shaw 8. icf. Taber 3.
98
SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
3. MATRICES AS QUADRATES.
423. Definition. A matrix (as understood here) is a quadrate of any order;
that is, a Sylvester algebra, usually of order >• 4°. Its units are called vids'
if they take the form
\j<s i, y = 1 n
424. Theorem. The general quadrate may be defined by the r ^= n~ units
Cab a, 6 = 1 n
such that
where
e„b = *■"/'
s=l
ji = ijij
in — jn ^ l^la jhyx — ^
2n
271
(J = COS 1- V — 1 sin —
n n
If
a, h= I . . . n
c, c£ = 1 . . . n
a,b, c,d ^ 1 . . . .n
425. Theorem.^ S .^ = x^ S .j'^ (-" ^ = x„,,
426. Theorem. Since every quadrate in the second form may be reduced
to the first form, it is easily seen that ^ satisfies the identity (characteristic
equation)
2(0'
(n-l)
2 (/<"-" iCsa 2 u"*"-" x,o — 4)
=
n— 1
in each term * 2 represents 2
8=0
427. Theorem. We may write
,^ = 2 i^f Sj-^' i-" ^
If a = 1 + t + i'^ + • • • -f i"~^, then the identical equation is
Saq> - <?) Sj-' a 4> ^y-<"-i' a ^
Saj^ Sj-'aj^-^ Sj- (»-» ay ^
Saj''-^^ Sj-^ aj"-' ^ /S;*-("-"ay<"-'' <?) — 4>
a, b:= . . . n — 1
a, b = . . . .n — 1
=
>Laguekre1; Cayi.etS; B. PeibceS; C. 8. Peirce, 4, 8 ; Stephanos 1 ; Tabeh 1 ; SiiawT; Lau-
BENT 1, 8, 3, 4. Ou the general topic see Bibliograplty of Quaternions.
'SUAW 7; Lauuent 1. 'Tabek I!.
«Cayi.ey3; Laoukuhe 1 ; Fuouenids 1, 2; VVethS; Tauek 1 ; Pascu 1 ; Bucuheim 3; MOLIEN 1 ;
Sylvestek 1 ; Siiaw 7; WniTEHEAD 1, and Bibliography of Quaternioni.
SYLVESTER ALGEBRAS 99
428. Theorem, (p may be resolved according to tlie preceding theorem
along any units of the form given by i,j, as
If J be such that
S.f^ = 0,J"=l 8 = l....n—l
then <p may be written in the form
(^ = a-yo + 2 a-,,,, i"j'' a=l....n — 1 h = . . . .n— 1
429. Theorem. Whatever number ^ is, /?' ^ /3~' has the same characteristic
equation as <p. Hence if this equation is
g" — W!iC"-i4- m.,^"-- — ± w„ =
not only is ^ a solution, but equally ^'^(3~'.
430. Definition. When
j"= 1 S.j'^=0 s = l n — 1
we shall call f^j-*=. K' .<p the tih conjugate of 4).
If <^ is in the form of §428,
A'' . <^ = a:oo + 2 x,,,, cj"' i'^f' a=l n — 1;h = 71 — I
Hence K' . ^ is the same function ^ of o' i that ^ is of i-
431. Theorem. We have at once
<p + K. ^ + K".^ + + K"-^ . ^ = 7uS<p = m^
(^ + /r. 4) + y = n\S'.<p = 'E..K''^K'^ s,f = n — 1
and since
(?)" + (A»- + = 4)- 4- 7r . 4)-- + = 7i.S.^'
therefore
21.K'^K'^ = n^ S~^ — 7iS^~=2m„ s,t = m — 1, s :^ t
Similar equations may be deduced easily for 3! m^j and the other
coefficients.
432. Theorem. If ^ . ct = (/cr, then
also if
(<?> - i/) ffi = (T, (?>— f/V'-'fTi^cTu {^ — gYoy = o
then
{K'^-g)f<y, =fa, .... (K'^ - g^-^ f a, =j%, {K'^-gYfa, =
433. Theorem. If the roots of <^ are such that each latent factor {<p — ^,)
occurs in the characteristic equation of <p to order unity only, then <^ may be
written
^ = ao + aii + of„_ii"-^
' Cf. Tabkr 3.
100
and
SYNOPSIS OF LINEAH ASSOCIATIVE ALGEBRA
^ . aj' = (fl„ + «! + ... a„_i) . aj'
-( («-ii
a„_i)faf
Hence the latent regions of K^^ are simply those of ^ transposed. This
does not necessarily hold when the latent factors enter the characteristic
equation to higher powers. We might equally say the roots of /f'^ are those
of (p transposed (cyclically).
434. Definition. The transverse of <p with respect to the ground defined
by i, j is
VJ-
It is evident that ^ = ^.
If ^9 = 1 we call ^ orthogonal. If <^ = <?> we call ^ symmetric or self-
transverse. If
<|) = 2 {Xij + V— 1 yij) ;iyo. (a;, y real)
and if
^ = 2 (a-ij — V— 1 yi,) ;iyo
then ^ is real if ^ ^ ^, unitary if ^^ = 1. hermitian if ^ = ^.
435. Theorem. The transverse of ^4' is 'i'^- Consequently <p^ =. <^^,
and ^<^ = ^^.
436. Theorem. We may write the equation of <p, if
<p = a + hi + ci' + +Jd"-^
So that
S . <?)— 4) S .i^ .... S . i"-~^
S.i'^ S .i^^ .... S.^ — ^
S.i-'-'^
S . i^
=
Tz^ = XK"^ K^^ a,b = n — 1, a:^b
Ts^ = 2/r"4) K^ip K'^
T„ 4) = ^A> A'-^) /f"-! ^
It follows that if the characteristic function of t be formed, it may be
written
^i--
or for
By differentiating this expression in situ the characteristic function for
^„ may be formed in terms of 4)1 ... . ^n. This function will vanish for
C, z= 9j ^„ = (|j„
^, = K'^,....^^ = K<^„
(iz= 1 . . . . n — 1 )
PEIRCE ALGEBRAS 101
XX. PEIRCE ALGEBRAS.
437. Ill the following lists of algebras, the canonical notation explained
above is used. In the author's opinion, it is the simplest method of expres-
sion. The subscrifjts only of the Jl will be given; thus (11 1) + a (122) means
Xju + aX,23. For convenient reference the characteristic equation is given.
The forms chosen as inequivalent are in many cases a matter of personal taste,
but an attempt has been made to base the types upon the defining equations
of the algebra. The designation of each algebra according to other writers*
is given.
The only algebra of this type of order one is the idempotent unit
ei = >7 = Xj,o=(110)
438. Order 2. Tijpe ^ {rj, i): {x — x^ e^f =
e2 = (l]0) ^^ = (111)
The product of ^ == a-jCj + XzCo, g = i/^ e^ + y., e.^ is
^a = ei {xi y.. + x., y^) + e^ [xo y^)
The algebra may be defined in terms of any two numbers t,, ^, if ^ ^^ 0,
so that we may put a in the form a = x^ + y^^.
439. Order 3. Type ^ {r„ i, i^) : {x — x^ e^f =
e3 = (ll0) e.,= {\ll) ei = (112)
The general product is
^a = cj (a-iys + x.jjz + x^yy) + e., {xny^ + %%) + e^ {x^s)
The algebra may be defined in terms of ^, ^', ^, if ^" :|: 0, ^ :|: 0.
Ti/pe * {y;, i,j): (x — a-g 63)2 =
eg = (110) -f (220) eg = (210) ei = (lll)
^0 = fj (itj //;, + x^yi) + 62 {Xi y-i + xg 2/2) + ^3 3:3 Vz = o^
The algebra is definable by any two numbers ^, a whose product does
not vanish. The product of ^a may be written
^a = GS^+^Sa — e3>SXSa
Hence
Also we may write the algebra (>;, ^', cr'), where ^', a' are nilpotents,
440. Order 4. Type^ {71,1, 1^,1^): (i — ar^ej^ =
ei = (110) 63 = (111) eo = (112) ei=(ll3)
If ^ = /S^ -(- V'(, then the algebra is defined by
^, C~, ^', K\ if ^4 + 0, V^^O, (F0^4:0, (70^:1:0
'Enumerations are given bj- PiNCHERLE 1 ; Catlet 8 ; Study ], 2, 3, 8 ; Scheffeks 1, 2, 3; Peirce3;
RoUR 1; Starkweatueu 1, 2; Hawkes 1, 3, 4.
-Study II; Scheffers II, ; Peirce a.,. 'Study III; Scheffers III, ; Peikce Oj.
< Study V; Scheffers IIIj. ' Study V ; Scheffers IV, ; Peirce a,.
7
102 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
Type {ri, i, j, f) : {x — x^ e^f =
e,= (110) + (220) 63= (210) + a (122) e., = (ill) + 6 (122) e^ = (112)
^a = — S^.Sa +aS^ + ^S(y + e^ {x.^ y-z + ax^i/g + b x.^y^)
or
F^ . To- = ^1 (3:3^2 + a a-3 2/3 + 6 x, ^3)
Hence
F^ . Va— Va . V^ = ^a — G^ = he^ (0-3 2/3 — 0:3^2)
We have two cases then : (l) when 6=0, (2) when J :J; 0.
We may determine e\=- e^, from
{V^f = e,{xl + a:4)
When a = 0, this gives us only one case of (t°=: e^.
When a ::j: 0, we may talce 4 = e^ as well as e| = e^ ; whence, if a ^
gg 63 = eg Co :=
If a -f 0, we may put « = 1
6363 = e.,€^—0
Finally, then, we have^
{vijf){\) f3 = (210) 63 = (111) 61= (112)
(>7*i.f)(2) ^3= (210) + (122) 6'2=(111) e, = (ll2)
{vijf){^) ^3 = (210) + (122) a e. = (1 1 1) + (122) e, = (112)
(>7Ur)(4) e3 = (210) ^3 = (111) + (122) ei = (112)
Type {yi, i, j, ij) : {x — x^ ej" =
e,= (ll0) + (220) 63 = (210) e. = (11 1) — (231) ei=(211)
^a = ^1 (.T3 y., — x.> y-i + Xg ?/, + a-^ y-^ + e. (a-, y^ + x^ ?/o)
+ ^3 (^3 ?/4 + a-4 7/3) + e^ Xi i/i
Defined ^ by ^^, <7, such that ( V^f = = ( Fa)'
Type'{-^,i,j,k): {x-x,e,f =
e,= (110) +(220) + (330) 63 = (210) ('3 = (310) ^ = (111)
V^Va =
Defined by any three independent numt)ers.
441. Order 5. 7)/pe ' (>7, i, r, i\ i*) : (x—x, e,f =
6, = (110) e4 = (lll) e3=(112)- ^.= (113) ei = {U4)
Definable by any number '( for which ( F^')' :^ 0.
'Stcdt IX is O7, i,J,j') (3) If «', = (310) -(111) + (c-l)(12ri), t, = (1U) + 2(132). Scuepfehs IV^ is
tUc 8am3. Peirce 6, and b\ reduce to this form. Studv X and ScuEt-PEUs IV^ reduce to (3); Study XI
and SoiiEKFEiis IV^ reduce Id (1); SoiiEi'i'Kits [Vj roduues to (4) if ;i = — 1, otherwise it reduces to (S).
'Stody XIV; SOBEFFEKS IV,; PEruCE d,. aSTODY XVI; Soueffkks IV,.
«3CUBFFERS V,; Peiuce a,.
PEIRCE ALGEBRAS 103
Type ' {ri, i, j, f, f) : (a; — Xj e^y =
ej = (ll0)4-(220) ei = (210) + a(l23)
e, = (lll) + /v(123) eo = (n2) ei = (113)
(1) /> :|i 0, we may lake 1 = 1.
(2) h = 0, we may take a = 1, or
(3) b = = a.
Type ' (>7, i, h y , r) • (a; — a-5 e.f =
65= (110) + (220) 64 = (210) + 6 (221) + c (122)
e3 = (lll) + (?(221) + e(122) e2 = (21l) ei = (112)
(1) e4 = (210) e3=(lll) + ci(22I)
(2) 64 = (210) e3=(lll)+(i(22l) + (122)
(3) ^4 = (210) + (122) e,= {\n)
(4) ^4 = (210) + (221) 63= (lll)-(221) + e(I22)
(5) 64= (210) 4- (122) 63= (lll) + cZ(221) + e(122)
Type 3 (>7, i, i?, j, f) : (x — x^ e.f =
^5 = (110) + (220) + (330) ^4 = (210) + (320)
eg=(310) 63= (111) ei = (112)
Tl/pe ' {r;, i, j, h, !r) : (x - x,e,f =
f5= (110) + (220) + (330) 64= (210) +« (122) + Z-(132)
e3 = (310) + c(l22) + (Z(l32) e.= (HI) + e (122) +/(132) ei=(ll2)
(1) 64 = (210) + (122) 63= (310) + (132) e, =(111) ei = (ll2)
(2) 64 = (210) eg = (310) + (132)
(3) 64 = (210) eg =(310)
(4) 64=(210) + (l22)-y(l32) e3 = (310) + 7(122) — (132) ^^=(111)
(5) ^4 = (210) + (122) — (132) C3 = (310) + (122)
(6) 64= (210)- (132) 63 =(310) + (132)
(7) e4 = (210) + (l+a-')(122) e3=(310)
(8) e4 = (210) + (122) e3=:(310) ej = (1 11) — 2 (122)
(9) e4 = (210) + (122) e3=(310)+ 2(122) — (132)
e3=(lll)— 2(122)
(10) e4 = (210) — (122) + (132) 63 = (310) — t (122) — (132)
e, = (lll)— 2((132)
'SCHEFFERS V, is in (1), «, = (210) + (123) — (112), e, = (111) + 2(123); Schefpers V, is (2); Schef-
FERS V, is in case (1), a = 0, c, = (210) — (112) ; Schefpers V, is (3); Peirce 6^ is in (1), j= (111) — (123),
fc = (n3), J = (113), m = (210) + (133) + (112); Pkirce c^ is in (1), j = (HI) - (123), * = (112), I = (113),
m = (210) + (112).
«SCHEFFEKS V,. is(l); e, = ( 1 1 1) + ?. (231), e^= (210); V,, is (3? with d = — 1; V„ is in (5); V„ is in
(3) or (4); Peirce d^ is in (5); e^ is in (4); /^ is in (1); ^5 is in (5); Aj is in (3); tj is in (1).
•ScHEPFBRS V||.; Peirce J.. * These are in order Schefpers V^, — Vj^.
104 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
Type {r„ i, j, h, I): {x— x^ eS~ =
(1)1 65=1(110) +(220) + (330) + (440) c, = (210) — (l3l)
e3=r(310) + (12l) 6.= (410) ei=(lll)
(2)' ^5= (110) + (220) + (330) + (440) e4=(210)
eg = (310) 62 =(410) ei=(lll)
442. Order 6. Type ^ {yj, i, r, i?, i\ i^) : {x — Xa e^f =
66= (110) 65= (111) 6, = (112) 63= (113) 6,= (ll4) 61 = (115)
Type ' {-r, i, j, f, f, f) : {x-x, e,f =
66 = (110)+ (220) 65= (210) + a (124) e^ = (11 1) + & (124)
63 = (112) 63= (113) 61 = (114)
(1) a=l=b 65= (210) + (124) 6, = (lll) + (124)
(2) a = 0,J = l e5=(210) 64 = (ill) + (124)
(3) a = 0=b 65 = (210) 64 = (111)
Type ' (57, i, J, ij, f, f) {x—x^ e^)' =
(1) 65= (210) + (122)+ 2\/^^(22l) 64 = (111) + (221)
(2) 65 = (210) 6, = (111) + 2(123)
(3) 6a = (210)+ (123) 6^ = (111) + 2 (123)
(4) 65 = (210) 6, = (111) + 6^(221)
(5) 65= (210) + (221) e4 = (lll)
(6) 65 = (210) + (123) 64=(111) + (^(221)
(7) 66= (210) +(221) 64= (111) +(123)
(8) 65 = (210) e4=(lll)
(9) 65 = (210) + (123) 64= (111)
10) 65 = (210) + (122) e4 = (lll) — (221) — 2(122)
11) 65 = (210) + (122) 64=(111) — (221)
12) 65 = (210) + (123) 64 = (111)— (221) — 2 (122)
13) 65 = (210) 64 = (111) — (221)— 2(122)
14) 65 = (210)+ 2(1 q: v/:i^)(22l) + 4\/^=T(l22)+ (123)
e4 = (lll) =F >/^^^(22l) + 2(1 ± \/^^)122
15) 65 = (210) + 2 V-^ (221) + (122) 64 = (1 1 1) + (221) + 2 (l 23)
16) 65= (210) + 4 (221) + (123) 64 = (1 1 1) + (221) + 2 (122)
17) 65 = (210) + 4 (221) 64 = (111) + (221) + 2(122)
18) 65= (210) + 4(221) + (123) 64 = (111) + 4 (122)
19) e5 = (210) — (?H — 1)(221) — i(??? + 1) (?/> — 3) (122)
^^^(111) + ^'^ 3 (221)+ 2(122)
'SCIIEFFERS V,j . 'SCHEFFERS V33. apEiKCEn,,. * Peiuce ()g is ( I ) ; Cg is (3).
•These arc In order Staukweathek 4, 8, 9, U, 13, 13, 14, 15, 16, I'J, 30, 31, 33, 33, 27, 29, 30, 33, 33.
Also Peirce aoj and h)„ are iu (4), nd^ in (5), z,. in (6), «/„ in (8), ae„ in (tt), ?«,. in (II).
PEIRCE ALGEBRAS jq^
(1) .', = (210) + (.320) e,=(.310) ., = (111) e, = (ll2) «, = (113)
(2) c'6=(210) + (320) + (133) e,= (,310) + ( 1 23) 6,= (ill)
e., = (\['l) .-, = (113)
(3) ^6 = (210) + (320) -f (133) e^ = (310) + (123) 6,= ( 1) 1) + 2 (123)
eo={n2) e, = (ll3)
(4) e, = (210) + (320) e, = (;310) e,= (1 1 1) + 2 (123)
Co = (112) e, = (ll3)
Tljpe ' (r, i, j, k, /r, P) (^ _ ^^^ ^^y ^
(1) ^, = (210) e, = (310) e, = (lll) e, = (112) .i = (ll3)
(2) e,=(210) + (123) e, = (310) e,,= (lll) e, = (ll2) ., = (113)
(3) .a=(210) e, = (310) e3 = (lll) + 2(123)
«3 = (ll-3) ei = (ll3)
(4) e, = (210)-(133) e, = (3I0) + (l23) e3 = (lll)
. e3=(ll2) e, = (1)3)
(^) ^«=(210) e, = (310) e3 = (lll) + 2(133)
^^=(112) e, = (ll3)
(6) e'5 = (210) + r/(l33) e, = (310) + (123) e3 = (lll)
^ ^ e, = (112) ei = (ll3)
(7) e, = (210) + (133) e, = (310) + (123) ^3 = (l 1 1) + 2(123)
^2=(112) ei = (ll3)
(8) e, = (210) + (133) + (123) e, = (310) + (123) .3 = (1 H) + 2(133)
^2= (112) ei = (ll3)
'^m iv, i,j, ij, f, if) C.3 = 1 (a; _ ^^e,f =
e« = (210) + ^(l_co)(22l)-|<.(l22) e, = (111) + c. (221)-^ (1-.,) (1 22)
e3=(21l) + i(l_,,)(222) e2 = (112) + .,r(222) e, = (212)
Tl/pe (>:, i, J, k, ik, V') (^ _ ^^ ^^y ^
(1) e5 = (210) e,= (310) + (132) ^3= (111) + 6(i22) + c (132)
62 = (211) ei = (112)
(2) e,= (210) + (l22) .^ = (310) + (132) ^3 = (HI) + 6(122) + c (132)
e2 = (21l) e, =(112)
(3) e6 = (210)+a(122) e^ = (310) + (122) + (132)
e3 = (lll) + 6(122) + c(l32) e. = (211) e, = (112)
(4) e5=(210) e,= (310) + (122) +(132) e, = (1 1 1) + i (122) + c(l32)
^3 = (2n) f,= (ll2)
(5 e6 = (210) + (l32) ,, = (310)-(122) e3=(lll) ,,= (211)
(6) e5 = (210) e, = (310) 6>3 = (lll) + (i22) e, = (211) e;=(ll2)
(7) ^, = (210) e, = (310) e3 = (lll) ., = (211) e, = (112)
'These are in order Starkweather 3, 5, 28, 10.
' These are in order Starkweatheu 1, 2, 0, 17, 18, 24 2.^ 26.
106
SYNOPSIS OF LINEAR ASSOCIATIVE ya,GEBRA
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Type{r,i,j,h,l,l')
f5 = (2l0)-a(132)
66= (210)
{X-
e^= (310) + a (122)
e2 = (lll) + a (122) + fl(132) + a(142)
eo = (111) + a (122) + a (132)
<'3=(111) + a(122) + a(142)
62= (111) + o(132)
62= (111) + a(142)
e2 = (lll)
e^=(310)
63= (111) + (122) + (132) + (142)
eo = (lll)+ (122) + (132)
e2 = (lll) + (122)
Co = (111)
a-eCs,
Type {yi, i, j, k, J, il)
e5=(210) — (231)
66 = (210)
{x.
Type{Yi, i,j, k, I, m)
e,= (310) + (22l)
e2 = (lll)+(221)+(23l) + (24l)
6. = (111)+ (221) + (231)
e.,= (111) + (221) + (241)
62= (111) +(221)
62 = (111) + (241)
e2 = (lll)
e^ = (3l0)
62= (111) + (221) + (231) + ^241)
62 = (111) + (221) + (231)
62= (111) + (221)
62= (111)
y = o
e3 = (410)
6, = (112)
63 = (410)
61 = (112)
•^6^6.
f3=(410)
61 = (211)
e3=(410)
6i = (21])
{x-
C6=(210)
64 = (310)
63 =(410)
62 = (510)
a*6 e^f =
6i = (lll)
SCHEFFERS ALGEBRAS 107
XXI. SCHEFFERS ALGEBRAS.
443. Tlie following liwls include algebras of order less than seven, with
more than one idempotcnt. lieducible algebras aie not included, nor are
reciprocal algebras both given.' 'J'he idempotents are >;; direct units t, j. . . .;
skew units e.
444. Order 3. Type" (>7i ; r,.. ; e.,^) {x — x.^ e^) {x — x-^ Cq) =
63 = (110) 6. = (220) ej = (210)
445. Order 4. 'Type ' {yj^ ; j?., i; ejj) (a; — a-;, e^) {x — x^ e^f =
^^ = (220) ey=(110) e. = (lll) ei = (2l0)
Type M>7i ; >72 ; Cai , e'21) {x — x-i Co) {x — X4 e^)) =
e^ = (110) e;j = (220) <?. = (210) ei = (21l)
Type ^ (>7j ; y/., ; % , e,.) (x — x.j «„) (x — x^ gy) = ^
e, = (110) 63 = (220) Co =(121) ej = (2ll)
446. Order 5. Type ^ {1^1 , i, i~; ■^,, e^^) (x — Xj ?„) (x — X; Cq)' =
6-6= (110) e, = (220) e,= (lll) e.., = {\\i) e^ = (211)
Type '' {y:i,i; ^i,j; eoi) (^ — x^ eo)^ (x — xg eo)2 =
65 =(110) ^^ = (220) e, = (111) e2 = (222) €^ = (211)
Type M>7i , * ; >72 ; ^^i , e^i) (x — Xi Co) (x — X5 ej- =
(l)e6=(110) e,= (220) e. = (lll) e, = (211) ^^ = (212)
(2)e5 = (110) 6i = (220)+(330) e,= (lll) e. = (21l) ei=(310)
(3) 65= (110)+(220) e,= (330) e3 = (210) £, = (111) ej = (311)
Type ^yiifi; n-i) ^12, e,^) (x — x^ e^) (x — x^ e^f =
(l)e6 = (110) e, = (220) e3=(122) e^ = (210) ^^=(112)
(2)e6=(ll0) e, = (220) e3=(122) &, = (211) ei=(112)
Type ^" {m ; >72 ; 4, 4', e^i") (x — x^ e,) (x — Xi Co) =
66= (110) ei = (220)+ (330)4- (440) e, = (211) e, = (310) £^ = (410)
Type " (>7j ; >7o ; e^o, el^, e'.J,) (x — x^ Co) (« — ^i Co) =
e6=(llO) e^ = (220) + (330) 63= (121) e. = (211) ei = (310)
Type ^2 ()7i J Yi.,; yi-y, e,^ , e,^) (x — X3 e,) (x — x^ e^) (x — x, ej =
65 = (110) ei=(220) 63 =(330) e»=(221) ^^ = (311)
Type '^ (>7i ; )73 ; yi-^; e.^, e-^) {x — X3 Cq) (x — x^ e^) (x — Xj e^) =
eg = (110) ei = (220) 63 = (330) 63= (211) ei = (321)
'For algebras of order seveu see Ha WKE3 4. "These are in order Scheffsks V,,, V„, V,, •
•StUDT IV; SCHEFFEKS III, . Hawkes (V)3„, 3,j, 1,.
• Studi VII; SCHEFFEKS IV,. 'These are in order ScHEFFEBS V„, V,,; HawkE3(V)3 3.
< Study XV; Scheffeus IV,. '»Scheffebs V„; Hawkes (V) 5.
tSTUDT XIII; SCHEFFEBS IV,. >' ScUEFFEKS V„ ; HaWKBS (V) 6.
•SCHEFFEBS V,; HaWKES(V)1,. " ScDEFFEKS V^.
'SCHEFFEBS V,; HaWKES (V) 4. "SCBBFFEBS V,.
108 SYNOPSIS OF LINEAR ASSOCIATIVE AX^GEBRA
447. Order 6. Type^ {r,^, i, t\ P; r.^; e^-^) {x — x^e^) {x—Xze^^=
eg = (110) e5 = (220) e^ = (221) e,= {222) e.= (223) .'j = (210)
Type 2 (>:i , ii , j\ , jl ; r,. ; e.^) {x — a-j ?„) (a- — x^ e^f =
(1) eg = (220) + (330) e,= {\W) e, = (320) 63^(221) e.= (222)
ei=(212)
(2) e, = (220) + (330) 65=1(110) e, = (320) + (232) e3=(221)
63= (222) ei=(212)
(3) 66= (220) + (330) e6=(110) e^ = (320) + a(232)
63 = (221) + (232) e. = (222) ej = (212)
(4) eg =(220) + (330) e5=(110) e, = (320) 63 = (221) + (232)
e3 = (222) ei = (212)
Type ^ (>:i , ii , j\ , ii jl ; n. ; e^i) {x — Xg eo) {x — x^ e^f =
66= (110) eg = (220) +(330) e^ = (221) — (331) 63 =(320) e. = (321)
e, = {2\\)
Type ^ {y:i,h, j\ , h ; r,.^ ; Cgi) (^ — ^^ ^0) (•» — -^'s <?o)" =
eg = (110) eg =(220) + (330) + (440) e,= (320) 63 =(420) 60= (221)
ei=(211)
Type ^ {rn, h , iV' ^2, h; (^21) i^— ^5 e^f {x — x^ e^f =
e6=(220) €, = {\\Q) 6, = (221) e^= {\\l) e,= (112), e, = (122)
Type ^ (J7i , I'l , ji ; 573 , i-i ; e.^) {x — x^ e^f [x — x^ e^f =
e6=(330) ^^ = (110) + (220) e4 = (331) e3=(210) e3=(lll) e, = (31l)
Type'' (>7, ti, ij; y-.^ ; e^^, gj,) [x — x^ ep) (x — Xg e^f —
(1) e6 = (llO) 65= (220)+ (330) + (440) e, = (221) + (430) e3=(222)
6, = (310) 61 = (410)
(2)e6 = (ll0) es= (220) + (330) +(440) ei = (22]) e3 = (222)
62= (310) 61 = (410)
Tyjje ^ ()7i , ^1 , q ; >72 ; ^12 , e^i) (x — Xg e,) (x — Xj Co)^ =
(1) eg =(330) + (440) e^ = (110) + (220) £',= (132) 63 = (310) e. = (lll)
6, = (112)
(2) e, = (142)
Type ^ {y;i,h, j\ ; >:,. ; Co, , ei^) (x — xg t-o) (x — xg co)'- =
(1)^6 = (110) 65 = (220) + (330) + (440) + (550) e, = (320) + (540)
63 = (221) 62 =(410) p, = (510)
(2) 64= (320)
UIawkes (VI) 1, 1. < Hawkes (VI) 1, 6. MIawkes (VI) 3, 1, 3, 2.
= In order Hawkes (VI) 1, 3, 1, 4, 1, 2, . ' Hawkes (VI) 2< 1. » Hawkes (VI) 4^ 1, 4, 3.
aiUwKES (VI) 1, 5. 'IlAWKES (VI) 3, 3. «IlAWKES(VI) 3^ 3; 3,4.
SCHEPFERS ALGEBRAS 109
Type ' (>:, , i, , y, ; r,., ; e.,, , e,„) (x — Xj t'u) {x — aij Cq)^ =
(1) e„ = (330) + (440) e, = (110)4-(220) + (550j ^', = (131) e^ = {^\0)
e., = (210) ei=:(lll)
(2) e, = (141) e^^l^ill)
^!'/JPe " (>7i . *i i >:3, *2 ; ''12, eiO (^ — a^6 eo)' (a^ — ^s eo)' =
(1) ee = (440) rv, = (110 + (220) + (330) e^ = {\i\) e3 = (441) e,= (i40j
e, = (14l)
(2)e„ = (440) e5 = (110) + (220) e4 = (lll) e,= (441) «?„ = (240)
ei = (:i41)
(3)e„ = (440) e5 = (110) + (220) + (330) e,=:(lll) 63^(441) ^, = (340)
e^ = (240j
Type ' (m , ''i ; r,., , u ; e,o , Cji) (x — x, e„)^ (x — x^ e^f =
(1) ^8= (330) + (440) e,= (110) + (220) e^ = (310) + (421)
63 = (131) + (240) e, = (441) €^ = (111)
(2) e3 = (240)
(3) e, = (310) 63= (131) + (240)
(4) e3=(240)
Type^ (>7i, i\ ; r,.,; e[., e^l, e'l^) (x— x^eo) (x — Xjeo)' =
(1) 66 = (440) 6, = (ll0) + (220) + (330) e, = {in) 63 = (340)
e, = (140) 61 = (141)
(2) 6e=(550) 65= (110) + (220) +(330) + (440) e4 = (lll) 63 = (150)
6, = (250) 61 = (151)
(3) 63 =(350)
ej = (450)
Type ^ iyii,h;yi2; e'n, ^{2 , eg,) (x — Xg Cq) (x - Xg eo)' =
(1) 66 = (440) + (550j 65 = (110) + (220) + (330) e^ = (530) 63 = (140)
fe,= (lll) 6, = (141)
(2) e3=(141)
61= (240)
Type ^ (>7i , J'l ; )72 ; ^21 , eii, eij) (x — x„ 60) (a: — Xg 60)' =
66 =(440) + (550) 65 = (110) + (220) + 330) 6,= (410) 63= (141)
62 = (111) 61= (530)
Type ' {r;i ; y;^ ; 6,'.,, e;{ , ei'.i', ej-J) (x — Xg 60) (x — Xg 69) =
e^ = (660) 65=(110)+ (220) + (330) + (440) + (550) e^ = (460) e^ = (360)
60 = (260) 61= (160)
' Hawkbs 4, 3, 4, 4. * Hawkes (VI) 5, 3, 7, ], 7, 3. « Hawkes (VI) 8, 3.
sHawkes (VI) 5, 1, 5, 3, 5, 4. ' Hawkes (VI) 8, 1, 8, 3. i Hawkes (VI) 9, .
« Hawkes (VI) 6, 1, 0, 2, G, 3, C, 4.
»
110 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
%)e 1 {ra ; >:2 i «i2 , «)2, Cjo', %) (a^ — JCs ^o) (a: — a^B ^o) =
^5= (110) +(220) +(330) + (440) 65= (550)+(GC0) e,= (4G0) e3=(360j
62= (260) ei= (510)
!%7e 2 (>7i ; >72 ; Cjo e,'^ ej e^j) (a; — Xg e^ {x — a:^ fo) =
e,= (440) + (550) + (660) e^ = (1 10) + (220)+ (330) e4=(630)
63 = (530) e. = (250) e^ = (140)
%9e ^ ()7i ; ii ; yi.; 573 ; e^o, Cig) (a: — a^^ eo) (a: — a^s ^o) (a; — arg e^f =
ee=(110) e, = (220) e, = (330) e3=(313) ^.= (323) Cj = (333)
^l/F^ * (>7i > *i ; ^"~'j ^s-> ^2] , ^23) (a: — a;4 Co) (x — x^ e,,) (a; — a-^ Co)' =
ee=(110) e, = (220) e,= (330) f3 = (212) e2=(232) ei = (33l)
%'e ^ ()7i, ii ; >73 ; >73 ; ^13, eo,) (a; — a-^ Cq) (a: — ^5 Cq) (a: — a'c Cq)" =
66 = (110) eg =(220) 64 = (330) 63 = (231) 6,= (312) ei=(332)
Ti/pe ^ (>7i , ii ; yjo] vja] e.i , 632) (a: — ^i «o) (a: — scr, Co) (a: — a-j fo)' =
e, = (110) e5=(220) e, = (330) e3=(12l) e2 = (23l) ei=(331)
Ty2>e • {y-i, ; r,.. ; ^3 ; e:2, ^is, ^ss) (a^ — ^i ^o) (a: — arg Co) (a: — a^e <°o) =
ee=(llO) 65= (220) e, = (330) + (440) e3=(31l) e.^ = {A20) e, = {Z2\)
Type ^ ()7i ; 773 ; r,^; e^. , e^^ , e^) {x — x^ e^ {x — x^ e^ (x — x^ e^ =
(1) ee = (ll0) e5=(220) e, = (330) e3=(312) e2=(23l) ei = (322)
(2) e3 = (211) 63 = (320) ^^=(311)
(3) e2 = (32l) ei = (31l)
Type ^ (>7i ; yi-i\ »?3 i «i2, ^12 , ^si) (a: — a-^ e^ (x — x^ e^ {x — x^ e^ =
65 = (110) 65 =(220) e, = (330) + (440) 63 = (420) 6.= (130) ei = (321)
Type 1" (»:i ; r,.; ris] e^,, e.^, 631) (x - Xj eo) (x — Xg e^) (x — Xj Co) =
e, = (ll0) eg =(220) e,= (330) 63 = (211) e2 = (13l) Cj = (321)
1 Hawkes (VI) 10^. ' Ha WKES (VI) 3, 3. « Hawkes (VI) 4, , 83 1, 9, 3.
SHaTVKES (VI) 11,. 6HAWKES (VI) 73. » Hawkes (VI) Sj .
3 Hawkes (VI) Is 3. ' Hawkes (VI) 3,. '"Hawkes (VI) 83.
< Hawkes (VI) 63.
CARTAN ALGEBRAS
111
XXn. CARTAN ALGEBRAS.
448. Quadrates. The unit.s in tliis case have been given.
Dedekind Algebras. These have been considered.
Order' 7. ei = (110)
«?5 = (330)
Order 8. Type g, x (>?, i)
e, = (120)
r„ = (l30)
e, = {2\0)
e, = (230)
Xt c,
I'^o-
X Xo
X.,
This is biquaternions.
Type" Q, + (>?, i) + e^
X^ Cq X
2 —
=
Xi 6i
1 ^0'
X^
X Xg
x^ eo — X
(Xfi Cq — x)- =
ei= (220)
(110), (120), (210), (220), (330), (331), (l3l), (231)
Order 12. Triquaternions.
Order 16. Quadriquaternions.
It is not a matter of much difficulty to work out many other cases, but
the attention of the writer has not been called to any other cases which have
been developed.
ISOHEFFBRS Q,.
SSCHEFFERS Qj, Q,.
eo
ei
'0
eo
Cl
ei
«!
— fo
PART lir. APPLICATIONS.
XXni. GEOMETRICAL.
449. The chief geometrical applications of linear associative algebras have
been in Quiilernions, Octonions, Triquateniions, and Alternate Numbers.
These will be sketched here very briefly, as the treatises on these subjects are
very complete and easily accessible. What is usually called vector analysis
may be found under these heads. There are two other algebras which find
geometrical application in a way which may be extended to all algebras.
These will be noticed immediately.'
450. Eqiiipollences. The algebra of ordinary complex numbers
has been applied to the plane. To each point {x, y) corresponds a number
2 = a; + Z/*^!- The analytic functions of z (say /(z) where df .%■=■ f {z) . dz)
represent all conformal transformations of the plane ; that is, if z traces any
figure Cj in the plane, /(z) traces a figure C, such that every point of Cj has
a corresponding point on C^ and conversely, and every angle in C^ has an
equal angle in Co and conversely.^
451. Equitangentials. The algebra
ej
has also been applied to the plane. The analytic functions of z represent the
equisegraental transformations of the plane, such that /(z) converts a figure
into a second figure which preserves all lengths.^ To 2= x + Cj ?/ corresponds
the line ^ cos a; + >; sin a; — y = 0.
452. Quaternions. Three applications of Quaternions have been made to
Geometry. In the Jirst the vector of a quaternion is identified with a vector
in space. The quotient or product of two such vectors is a quaternion whose
axis is at right angles to the given vectors. Every quaternion may be
expressed as the quotient of two vectors.
' See Bibliography of Quaternio7is. Also the works of Ha.miltox, Cliffokd, Combebiac, Gkassmanx,
GiBBS and their successors.
'BeLLAVITIS 1-16 ; SCHEFFERS 10. ' SCHEFFERS 10.
113
e.
ei
<'o
%
ei
ei
ei
114 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
The following formulae are easily found :
(1) If a is parallel to (3 F . a/3 =
(2) If a is perpendicular to /3 S . a^ =
(3) The plane through the extremity of ^, and perpendicular
to a is iS{p — b) a =
(4) The line through the extremity of a, parallel to jS F(p — a) jS =
(5) Equation of collinearity of a, /?, 7 V{a — /?) {^ — y) =
(6) Equation of coplanarity of a, /3, 7, ^ S{a — ^) {(3 — y) (7 — ^) =
(7) Equation of concyclicity of
a, /?, y, ^ y{o^ - 1^) (^ - r) (y-^) (5-a) = o
(8) Equation of cosphericity
of a, 13, r,h,s S{a-P) (/3-y) [y-h) (5 - e) {s-a) =
(9) The operator q{)q~^ turns the operand () through the angle which is
twice the angle of q, about the axis of g-. The operand may be any expression,
and thus turns like a rigid body. These operators give the group of all
rotations.^
(10) The central quadric may be written /5p<?)p = — 1 =■ gf -\- 2 iS?.p S^p,
where 4) is a linear vector self-transverse function ; /I and fi are the cyclic
normals;
a = ^gz — gi i + *</gz—g2 /^ 2^ V = ^9^ — 9i *"— '^93 — 9z^
i and 7c being in the direction of the greatest and the least axes, and the axes
are given by </i = %- , gi = j^ , 93 = —tt- Conjugate diameters are given
by Sa^^ = S^S^y = Sy^a = 0.
(11) For any curve, p = <^(^), any surface, p = ^(/, ?() or F{p) = {).
dp is parallel to the tangent of a curve, ^ 7 //" ^^ ^^^^ vector curvii-
ture, Udp S y'/^ri is the vector torsion, a = Udp is the unit tangent,
^ =. JJVdpd^p Udp is a unit on the principal normal, y = UVdf d"p is a unit
on the binormal. Forasurface i^(p)=0, yi^ is the normal, S{p—p^ v^o =
is the tangent plane.*
The second application^ of quaternions to geometry is by a homogeneous
method. In this the quaternion q is written q = Sq{l + p), and q is regarded
as the affix of the point p with a weight Sq = w.
1 Catlev 10. 'Hamilton's works, Tait's works, Joi.y's works.
'This applicatioa may be followed in Joly 30, 11, 35; Shaw 3; Chapman 4; see also Bkij,l 1.
GEOMETRICAL
We write also
A . q Ars -=■
r
Sqr
8
Sqs-
8
t 11
A . qr Astu =^
Sqs
Sqt Squ
Srs
Srt Sru
t
U V
A . qrs A . tuvw =
Sqt
Sit
S7/11 Sqv
SJ-ii Srv
Sst
Sm
Siv
115
w
Sqio
Srw
Saw
In pai'ticular we may write
— A.l Aah = A' . ah
A . ah Aijh — A". ab = V . Va Vh
S . a Abe Aijk =S .VaVb Vc = SA . abc
A . abc := — K . A . abc A 1 ijh
S . a A . bed = — Sd Abed A . 1 ijk
We have
(1) The equation of line a, b is A . abq = 0.
(2) The equation of plane a, h, g\s S . q Aabc =^ 0.
(3) a, b and c, d intersect if S . a Abvd^= 0.
(4) The point of intersection oi S .lq-=-0 =■ Smq = Snq is q=- A . Imn.
The third application of quaternions is to four-dimensional space. ^
(1) Any quaternion p represents a four-dimensional vector in parabolic
space. All vectors parallel to p, in the same sense, and equal in length are
represented by p.
(2) If g^ is a second vector, then the angle Z {p, q) being Q
cos Q = SUp U^ = S.llJiUq
(3) The condition that^ is perpendicular to q is Spq =■ Spq = 0.
(4) There is for p as a multiplier /> () a system of invariant planes, one
througli any given line q, called a system of in-parallel planes. Multiplication
by p, p, has also a system of invariant planes, called by-iKirallel plants, one
through each line q. The displacement of (7 in any invariant plane is constant
and equal to the angle of/). The tensor of (7 is multiplied by the tensor oi p.
If 3' is resolved parallel to two invariant planes of /), these components turn in
their planes through Z p, and the product pq has these results for its
components.
(5) If Vqp = 0, g- is parallel to p.
(6) The projection of q on }) is Up Sq KZTp.
The projection of q on a vector perpendicular to p is Yq KUp . Up.
(7) The plane through the origin and the two vectors from the origin
a, — (X2 and ai («! — a^) is aip -\- pa^ =
1 Hatha WAV 2, 3, 4, 5; Stuingham 4, 5, 7.
116 SYNOPSIS OP LINEAR ASSOCIATIVE ALGEBRA
The plane through the point aj a containing the vectors
ttj — Uo and tti (ttj — tto) is aj g + quo + 2a =
(8) If "
ai = ± UVec, ao=:± f7Fec, and a = — aj Oq
then the equation of the plane through Uq containing the vectors c, e is
ttjjp 4-^Jao + 2a ^
(9) The plane through c, d, e is given by the same equation with
«! = UV{cd + d~e-\- ec) a. = UV{dd + de + ec)
a ^ — M"i ^ "I" ^'^2) ^ — i («! <Z + c^ag) := — ^ («! e + eag)
(10) The normal to the plane is a^a.
(11) The point of intersection of the two planes
ttip +i)a2 + 2a = = Pii^+p^o H- 26
is
/?! a — a^So + ai 6 — fcag
-^ S{a2^2 — ^i/iJi)
(12) If the two planes through the origin (aj ag 0) {^i^^Q) meet in a
line through the origin, it is necessary and sufficient that
The cosine of the dihedral angle between the planes is ± Sa^^i = ± Sa..^^-
They are perpendicular when this vanishes.
(13) The two planes {ayaz^a) {^i[^o2b) meet in a straight line if
ai 6 — 6 tto + /3i a — a /^a =
Let
/=aiZ) — Sag g=-^ia — a^2. ''^ = '^{(^2^3 — ai/?i)
then if /= — S' ^ 0, the equation of this line is
X — 2 Vab
t=—f—
(14) The two planes meet in a point at infinity if wj = and /+ </ ij: ;
they meet in a line at infinity if
/6?i = ± ai /i?2 = ± ao
(15) The perpendicular distance between the planes (aia3 2a) (aj a, 26)
is in magnitude and direction aj (a — 6).
(16) The vector normal from the extremity of c to the plane (aiajZi) is
^ ttj ( 2a + «i c + Ctto)
(17) The vector normal from the origin to the intersection of
oh — ha . , ,
(aia2 2a) and (/3]^2 26)is y + r/ = 0, /4:gr
OOC1 — fXo
(18) Two planes meet in general in a point or in a straight line. Through
any common point transversal planes may be passed meeting the two in two
GKOMBTRICAL 1 1 7
straight lines m, v and forming with them equal opposite interior dihedral
angles. The angle between these lines u, v is the isoclinal angle of the two
planes. Two planes have maximal and minimal isoclinal angles if there exist
solutions c-\-u\ti and c + (Sfv of their equations such that
)Sai uv = S^i MW = Suv df: Sa^ u^^v
The planes of these angles and these only cut the given planes orthogonally.
The lines u and v are given by
« = «! (yi + nO — {yi + 72) «3 yi = UVrxi (3
v = ^i iyi + r,) - (/I + y-z) (^2 n = ^^«3 /?i
«i " = «' = «i (ri — r^) — (/i ~ /z) «2
^iV = v' = (3i{yi — /») — iyi — /a) (3^
(19) There are no maximal and minimal isoclinal angles if any one of
the four conditions is satisfied :
/?! = ± ttj /So = ± a2
In this case the isoclinal angle is constant for all variations of 0.
(20) Two planes are perpendicular and meet in a point if
Sai /?i = — Sa^^z 4: or :}: 1
Two planes are perpendicular and meet in a line if
Two planes are hyperpendicular if every line in one is perpendicular to every
line in the other. In this case
^ai /3i = — /Sixg /?2 = ± 1
that is
/?j := ± tti /?o =: ^ a..
If two planes are parallel
Ui = /?i ag ^ /?3
453. Octonions. The following are the simpler results :
(1) The vector from to P is a rotor p and may be transferred anywhere
along its own line. It is not equal to any parallel rotor. Rotors from the
same point O are added like vectors, p + e being the diagonal of the parallelo-
gram whose sides /7wn are p and e.
(2) The side parallel to p is p -f flil/ep, that parallel to e is s + HJ/pe.
(3) If all vectors are drawn from 0, the usual formulae of quaternions
hold. Thus the equation of the plane perpendicular to 6 through its extremity
is S {p — (S) f^ = ; the line through the extremity of 8 parallel to a is
p := h -\- ta. But a rotor in tlie plane is not p — 6 but p — b -\- D.M . hp and
a rotor on the line is not xa but x (a + HJ/rW).
(4) A velocity of rotation about an axis is represented by a rotor on
that axis, a translation along the axis is a lator on that axis. A motor, as
G) -\- HcT, indicates a displacement such that in time dt any point rotates about
the axis of the motor by an angle 71) . dt and is translated along the axis by
a distance Tadt.
8
118 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
(5) The axis of M . AB is the common perpendicular of the axes of A
and B. The rotor of J/. AB is the vector of the product of the rotors of A
and B considered as vectors through 0. The lator of M. AB has a pitch equal
to the sum of the pitches of A and B and the length of the common perpen-
dicular multiplied b}' the cotangent of the angle between A and B (= d cot 0).
(6) The rotor of J. -f -B is equal and parallel to aj -|- /3, the sum of two
rotors from equal to the rotor of A and parallel to the rotor of B respect-
ively, and intersects the common perpendicular from J. to i? at a distance
from equal to [w being the common perpendicular]
T \m SI3 {a, + 13)-' + {p~ p') Ma, /?.(«! + /?)-==]
(7) S, . ABC is one-sixth of the volume of the parallelopiped whose edges
are the rotors of ABC. M, . ABC is a rotor determined from the rotors of
A, B, C &s V . a^y is from a, /?, y.
t . S . ABC = tA->rtB-\-tG+ dcotd — etan^); d and as in (5), e and ^
the common perpendicular from M . AB to C, and the angle.
,_,„„ A , .r, , .^ dcotd — etand)
t.MABG=tA+ tB + tC — ^Towi — o—-; ,-a ,+ — ^
cot-t/tan-^ -f cot-y + tan-^
(8) If 5 and C are motors whose rotors are not zero and not parallel,
then ^B -\- YG is any motor which intersects the common perpendicular of
B and C perpendicularly.^
454. Triquaternions. If /<, ^i' are points, h, b' lines, w and ro' planes, all of
unit tensor,
{i = ^j-o + (0 {ixi + jxo + Z'a-g) [I is the point -^- , ^ , -f
a = (j/?„ + ^ (iaj + joi2 + /I'ttg) a is the plane /?o ^o + «i ^i + «2 ^^s + ^3 a^s =
b =iai+ jao + kas + o {ii^i + j^.^ + ^^^s) «i Pi + <^z P-z + "s /^3 =
h is the line 1^ ., P^ ., Pl^^ , 1^ , J;^. ., P^
tti ttg as /:Ji /^3 /:^3
That is, a point or a plane is represented by the symmetry transformation
it produces; a line, by a rotation about it as an axis through 180°.
(1) Ghh' is — cos of angle between the lines.
(2) Gmm' is — cos of angle between the planes.
(3) L(i^' is the vector of n' towards {x.
(4) L^h is the vector perpendicular of the plane containing the point and
the line, tensor equal to distance from point to line.
(5) Lhh' is the complex whose axis is the common perpendicular and
whose automoment is tlie product of the shortest distance by the
cotangent of the angle.
(6) Liim is the perpendicular drawn through ^ to the plane m.
(7) Lhm is the point of intersection of the line and the plane, tensor equal
to the sine of the angle of the line and plane.
'M'AULAT 2.
GEOMETRICAL 1 ] 9
(8) Lmm' is the line of intersection of the two planes, tensor equal to the
sine of the angle.
(9) P^ih is the plane through ^ perpendicular to h.
(10) Phh' is the plane at oo multiplied by the shortest distance and the sine
of the angle of the two lines. 7hh' is the moment of the two lines.
(11) P^m is the plane at ro multiplied by the distance from fi to m and
positive or negative as ^ is on the side of the positive or negative
aspect of the plane.
(12) P8m is the plane drawn through the line perpendicular to the plane,
tensor equal to the sine of the angle of the line and the positive
normal of the plane.
(13) If y, y' are two complexes of unit tensors, Vyy' =: means the two
are in involution.
(14) A displacement without deformation is given by r()r~^:
r = q + aqi Sqqi = P [r' — (Xr)^] =
The axis is 5 = VLr = U{Vq + a Vq^).
a
The angle of rotation is 2d. 6 = tan~^ tv~
The translation is 2yj. ri
_ Sq,
TVq
r = (1 + cjrV) (cos + 5 sin 6)
(15) Transformations by similitude are given by j- = fiq -\- aq^. Sqqi =
(16) The triquaternion ?• produces a point transformation m' = rmr~^,
if r = ?u + 7 + p, 2wp ~.FP=0
This transformation may be written ^ -, which is a
rotation about the line d, and a homothetic transformation whose
center is OT and coefficient i^ rr •
w -\- Im
Hence r produces the group of transformations by similitude.'
(17) A sphere^ is represented by the inversion which it leaves invariant;
that is, by the quadriquaternion ^ (z'xj -^-jiji + Icz^ + uirj + u'w^.
(18) If J/ and M' are two spheres of zero radius, m and m' their centers,
L m M' ■=■ L m' M is the line {mm'). The sphere on w? m' as diameter is
Pm M'. If cZ is a line, then P . Md is the plane through d and m.
455. Alternates. Tliere are various applications of the different systems
of alternates, notably those which are called space-analysis — the development
of Grassmann's systems; vector-analysis — a Grassmann system without the
use of point-symbols or else a system due to GiBBS; and finally the Clifford
systems. No brief account or exhibition of formulae can be given.^
'COMBEBIACa. 'COMBBBIAO 3.
'See Bibliography of Quaternions; notably Jolt 6 ; Htde 4 ; Whitehead!; Gibbs-Wilson 3.
120 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
XXIV. PHYSICO-MECHANICAL APPLICATIONS.
456. These are so numerous that they may be only ghmced at. Quater-
nions has been applied to all branches of mechanics and physics, biquaternions
and triquaternions to certain parts of mechanics and physics, alternates and
vector analysis in general to mechanics and physics. The standard treatises
already mentioned may be consulted.
XXV. TRANSFORMATION GROUPS.^
457. Theorem. To every linear associative algebra containing a modulus
belongs a simply transitive group of linear homogeneous transformations, in
whose finite equations the parameters appear linearly and homogeneously, and
conversely."
458. Theorem. Associated with every linear associative algebra containing
a modulus and of order r, is a pair of reciprocal simply transitive linear
homogeneous groups in r variables.^
459. Theorem. To a simply transitive bilinear group which has the
equations
XJ= 2 o.M.Xu -^— (t = 1 . . . . r)
s
corresponds the algebra whose multiplication table is €^6/,=: "S. a,^, e^, and
s
conversely.*
r r
460. Theorem. The product of a = 2 aiCj and h =.1,h^ei gives the finite
transformation corresponding to the successive transformations^ of the para-
meters («!.... Or) and (Z>i .... h^).
461. Theorem. To every sub-group of G, the group corresponding to the
algebra 2, corresponds a sub-algebra of 2, and conversely. To every invariant
sub-algebra of 2 corresponds an invariant sub-group * of G.
462. Theorem. To the nilpotent sub-algebra of 2 corresponds a sub-group
of (r, Fi (/)... . Yk (/), such that for no values of Y ./or X ./, transforma-
tions respectively of the sub-group and the group, do we have**
Y{X/) = c,Xf c. to
X{ Yf) = JXf <y 1
( FX) = Y{Xf) - X{ Yf) = <.A7 c. t
463. Theorem. The invariant sub-group g, corresponding to the nilpotent
sub-algebra a, is of rank zero.*
'Stddt7. 'Poinoare 1, 2, 3; Study 8; Cautan 3. See also Schur 1.
3Studt1,3; Lie-Sciieffeks 4; Cautan 3. *Cartan 3. 'Cautan 3; Study 3.
'Cautan 3. Cf. Engel, Kleincic Beitriige zur Gruppentheorie, Lcipziger Berichte, 1887, S. 'J6; 18',)3,
8. 360-369.
TRANSFORMATION GROUPS 121
464. Theorem. To every quadrate of order r ■=■]>" corresponds the para-
meter group of tlie linear homogeneous group o\' £) variables.*
465. Theorem. To every Scheffers or Peiuce algebra corresponds an
integrable simply transitive bilinear group, whose infinitesimal transforma-
tions are
o -a
Xi = Xi 23- + - 2/p 3y * = /3p («( , ^i are the characters of r^i)
and whose finite equations are^
2/1 = <% Vi + ^i a-.. + 2 a,^, h^ y, (>. < i, /t/ < i j
A, (*
466. Theorem. Every simply transitive group can be deduced from a
group of the form just given,
y» = ^'"i^-gfar + 2 a,,, r^^ -g^(.r («> ^ s>J, A=«o «»=«i, /3.=/?0
or
by setting to correspond to each variable A''*"' or F'*"^ of character (a/3), p^P^
new variables a'^', y'-^\ where i,j are respectively any two numbers of the
series 1, 2 .... p„, 1, 2 .... ^)^. Likewise to each parameter J."'', JS'"' of
character (a/?), Palh new parameters a|j', h'^\
The simply transitive group is then defined by the infinitesimal trans-
formations
X% = 2 xt -^ +" 2"'>i -g4- (/?P = i; a, ^ = 1, L> . . . . i..)
Pi g 8
or by the finite equations"
1,2. ...Pi
A
«/(i) V „(^.) ,,,(t) I 5" ft(i) -(<■.) _|_ y „ ^C") ,,(f)
2/ a? — "A3 yaA T^ -i ^\^ ""^aX T^ ^ CCp^j O^p ^^
A A fwA
'CARTAN2; MoLiEN 1. Cf . Cayle Y 11, 5 ; Laouerre 1 ; Stephanos 1 ; Kleis 1 ; Lipschitz 2. Also
Cayley 3 ; Frobenius 1; Sylvester 1 ; Weyr 5, 6, 7, 8.
« Cartan 3.
122 SYNOPSIS OF LINEAK ASSOCIATIVE AX.GEBRA
467. Theorem. Every simply transitive bilinear group G is formed of a
sub-group r of rank zero^ and a sub-group g which is composed of h groups
(Ji • ■ • • 9hf respectively isomorphic with general linear homogeneous groups of
Ih) Pz • ■ • • Ph variables. Moreover the variables may be so chosen that the
p\ first variables are interchanged by the first gi of these h groups, like the
parameters of the general linear homogeneous group on p^ variables, and are
not altered by the other h — 1 groups ; the same is true of the p\ • ■ ■ ■ p\
following variables ; finally these p\+ . ■ • ■ -^ p\ variables are not changed
by the sub-group ^ V.
468. Theorem. All simply transitive groups are known when those in
§ 465 are known."
469. Theorem. Every real simply transitive group G is composed of an
invariant sub-group V of rank zero, and a sub-group g which is the sum of h
groups gi ■ ■ ■ ■ Qh, each of which, belongs to one or other of the three types
following:
(1) The groups of the first type are on p? variables xy and are given by
the formulae
^V - ^li ^y.^, -+- ^zi g^^^ -1- • • • • + a-pf 2^^^
or
•*• ij -~ ^U •'^il T ^2j "^12 "T • • • • I (tpj ^ip
giving the parameter group of the general linear homogeneous group on p
variables.
(2) The groups of the second type are on the 2p^ variables Xij, y^ and are
given by the formulae
^ij - o^u Q^^,+ ■■■■ + Xpi a^. + 2/ii a^ + • • • • + Vpi dy^^
^--^"3^,,+ •••• + ^''^ 3^,, - 2^1^ Bx,, + ••••-y-*ax,,
or
X ij — (iij Xfi -\- ■ ■ ■ ■ + cij,j ^ij, — 6]j ?/ii .... — bpj yip
y'ij = «]; 2/ii + + Cpj !/ip + h a^ii + + hj ^iP
(3) Those of the third type are on 4p^ variables Xy, i/ij, 2,j, t^j, given by
the formulae
\j <^y\j ^*Aj WAj
3 3 .. 3 , . 3
<^!/\j ^'^/^j '"-A^- ^^>~j
3 3,3
Y,j-^^^(x,, 3,„ -2/A. a^--^^i:a^ + ^^* dzJ
y _i ( 3, a a ,3\
^'^ - i K""'' 3^ + ^« -3^:7 - '^* 3^ ~ ^'' 3^,, ;
„, '.' / 3 3 , 3 , 3 \
iCartan3. Cf. Molien 1. «Cabtan:
TRANSFOUMATION GROUPS 1 23
or p
x'y = i; {a^j JCu — ^Aj y.A — fh} Z(A — f^A^ <<a)
A = l
V
y'ij = 1 (a, J yt^ + h^^ Xo, — c^^ ti^ — d^j z,J
A = l
V
A=l
P
fij = 2 (a,,- ti, — b,j Zi^ + 6\; ?/„+ (Z,,- a-(,)
A = l
To each of these groups o£p^, 2p^, or 4^/ variables we can set to corre-
spond p^, 2p^, or 4^/ variables which are interchanged by these equations,
without being changed by the other groups which enter g nor by the sub-
group r. All these variables are independent.^
470. Theorem. The groups in §469 are not simple, but are composed of
an invariant sub-group of one parameter and simple invariant sub-groups of
p^ — 1, 2p" — 1, 4jr — 1 parameters.^
471. Theorem. Simply transitive bilinear groups in involution (transfor-
mations commutative) are given by the formulte
(1) ^=x3^-f22/./- ^'="4. +5"^"^^'!:
i=l, 2 r—1 oL^is = if s^i, 8^2.
or
x' = ax y'i = a!/L + hiX+ 1 a,^; b^ y^
Kv-
3x
^ P\ 0\ o
or^
a;' =. ax — cz
z' ■=.az -\- ex
y'i = ay^— ct^ + liX ~ diZ -\- 1 a,^i (6^ y, — d^ Q — 1 13,^^ {b^ t, + d^ y,)
t'i = at, + cy^ -f b, z + d, .t + S a,,^ {b, t, -f d, y,) -f 2 /?.,.■ [b, y, - d, Q
472. Theorem. Every bilinear group G is composed of an invariant sub-
group r of rank zero, and of a sub-group g which is the sum of a certain
number h of groups which are respectively isomorphic with the general linear
• Cabtan 2.
124 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
homogeneous group onp^, p^ ■ ■ ■ • Th variables. Every real bilinear group G
is composed of a real invariant sub-group V, and a sub-group g which is the
sum of h groups each isomorphic with one of the three following groups:
(1) The general linear homogeneous group on ^ variables.
(2) The group on 2jf parameters and 2p variables a:,-, yi'.
x\ = ciii xi + + Opi a-p — &i;J/i — — ipi l/p
2/i = «i(2/i + + «pi Z/p + ^li a-i + + bpi Xp
(3) The group on 4p^ parameters and 4p variables a-^, ?/;, Zj, i^:
p
x'i = 2 (a^i x^ — i,i y^ — c^( z^ — d^i t^)
P
y\ — 2 {a>,^ y^ + ft^^ x^ + c^^ t^ + d^ zj
A=l
V
z'i = S [a^i z^ + Z>^j t^ + c^i x^-\- d^i y^)
A=l
V
t\ = 2 (a^i t^ + 6^i z^ + c„. ?/^-f rf;^, xj
Each of these groups is formed of a simple invariant sub-group on p^ — 1,
2p~ — 1, or Ap? — 1 parameters and an invariant sub-group on one parameter.^
473. Theorem. Every bilinear group G is composed of an invariant sub-
group r of rank zero, and one or more groups g^, Qn . . ■ . of which each g is,
symbolically, the general linear homogeneous group of a certain number of
variables X^ . . . . X^, these variables being real, imaginary, or quaternions,
and the p^ parameters having the same nature,
X'(p=2X«A.
A = l
If the variables and the parameters of the bilinear group G are any
imaginary quantities whatever, the group is composed of an invariant sub-
group r, of rank zero, and of one or more sub-groups g^, g^ ■ ■ ■ ■ of which
each g is the general linear homogeneous group of a certain number of series
of p variables, of course imaginary.^
474. Theorem. The quaternion algebra is isomorphic with the group of
rotations about a fixed j^oint,^ with the group of projective transformations on
a line, and with the group of special linear transformations around a point in
a plane.
475. Theorem. Biquaternions is isomorphic with the group of displace-
ments in space without deformation.^
476. Theorem. Triquaternions is isomorphic with the group of displace-
ments and transformations by similitude.* Quadriquaternions is isomorphic
with the group of conformal transformations of space.
' OAETAN 2. 'CAYI.EY 10; I.AOUKKllE 1 ; STEPHANOS 1 ; SxitlNdllAM S; BkezI.
'M'AULAT 2; COMBEBHC I ; Stl'dt 5. * Combebiac 2, 3.
AnSTIlACT GROUPS | 25
XXVI. ABSTRACT GROUPS.
477. Theorem. Every abstract group is isomorphic with a Fkobenius
algebra of the same order as the group.'
478. Theorem. Tiie expressions for the numbers of tlie Frobenius algebra
corresponding to the group are determined by finding the sub-algebra consist-
ing of all nMmi)ers commutative with every number of the algebra, then
determining by linear expressions the partial moduli of the separate quadrates
of the algebra, and then multiplying on the right and on the left by these
partial moduli. Every number is thus separated into the parts that belong to
the different quadrates. The parts for any quadrate of order r^ determine
the rf quadrate units of the sub-algebra consisting of the quadrate, which
p
determination is not unique. In terms of these r = t ij units all numbers of
i = l
the algebra may be expressed."
479. Theorem. The characteristic equation of a Frobenius algebra con-
sisting of 2' quadrates is the product of ;j irreducible determinant factors. The
prelatent equation and the post-latent equation are identical and consist of the
products of these p irreducible factors each to a power r^ equal to its order.^
480. Theorem. The linear factors of a Frobenius algebra correspond to
numbers which are commutative with all numbers. The number of linear
factors is the order of the quotient-group; that is, the order r divided by the
order of the commutator sub-group.
481. Theorem. The single unit in each of the quadrates of order unity,
may be found as one of the solutions, cr, of the equations
^(y = a^=ta for alU"'s
For the ^'s it is sufficient to take the r numbers corresponding to the operators
of the group. Thus if cr = 2 . a-^ C;, and if e, Cj = e^j, hence <=•,• e,-i^ = e,, we
must have
ta -=0 . ej = I, . Xi Cij = 2 x^ - 1 e^ for all y s
Hence
tXi = Xij~l
If ej is of order (ij, ep = fi = 1, then
Xij-i = tXi Xij-2 = f Xi .... Xfj-a = t'Xi fj = 1, or a:, =
Hence
27tn , , — - . 27tn
t = cos \-V—l sin = L n = 1 . . . . w,
since not all a-j vanish.
»Poincark4; Shaw 6. This theorem follows at once from Cartan 2. See also §121.
>Poincare4; Shaw 6. 3 Frobenius 14 ; Suaw 6.
1 26 SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA
a = -Exi{ei + tj eij-1 + {^Cij-z + . . . . )
= ^Xie^{l + (jej-i+ .... +ip-'ej) {j=l....r)
The subscripts i run through those values only which are given by the
table \ G\ = {eiCj]. By operating on <y with other numbers e^^ we establish
other equalities among the a;'s and finally arrive at the units in question.
482. Theorem. The units in the quadrates of order 2^ may be found as the
solutions of the equations
^~<y = ti^a + f^o o'Q =: <i cr^ + <3 (7 (^ any number)
We may state this also
(?i ?2 + ^2 ^i) o' — ^ ^1 (7 — ^2 ^2 cr + /3<^ = (^1, ^2 aij numbers)
The units in the quadrates of higher orders may be found by similar
equations.
483. Theorem. The numbers e^, i = 1 . . . . ?•, may be arranged in con-
jugate classes, the sum of all of those in any class being commutative with
all numbers of the algebi'a. If these sums are K^, K^ . . . . K,^^ then
n
i=l
The partial moduli of quadrates of order 1", are formed by operating on a
with all numbers and determining the coefficients to satisfy the equations
^(T :=ta cr^ = ta
The partial moduli of quadrates of order 2^ and higher orders satisfy the
equations of §48 2.
484. Theorem. Every Abelian group of order « defines the Feobenius
algebra ^
Cf = ;iiio (i = 1 r)
485. Theorem. The dihedron groups, generated by gj, 63,
6j™ = l=e| €361 = 61™-! 63
define Frobenius algebras as follows :
When m is odd : Let (j™ = 1, u being a primitive root of unity, then the
algebra is given by
/"-no ■^220 '^2i — 1> 2i-l> '^^2i — 1> 2i> ■^2£>2i — 1>0 ^^21 1 3( > (' — A.... ^ )
We notice that
Cj ^ ?.j]o + ?-320 + ^ (<■* * '^i.'i + 1 ) 2J + 1 > + '■' '^21 +2> 2i + 2) o) ( i = 1 .... ^ J
Cg = /IjiQ /I220 + 2 (^21 + 1 > 2i + 2 > "i" ^^21 + 2 > 2i + 1 > 0) ^t = 1 .... - J
' Shaw 0. This reference applies to Uie following sections.
ABSTRACT GROUPS 127
Wlien m i.s even, the algebra i.s yiven hy
^WQ ^"Z'iO ^Wa ^'<40 '^2(-l} 21- 1) '^■2i—\}'il}0 ''•aO2i-l>0 '^-ZltZiid
We notice tlual '
^1 ^ '^110 + '^XiO ^zm /'.410 + i <■> ~ '''•2l-l> 2(-l> ^
+ :i-"-^'>.2„2„0 (i=3....^-+l)
*2 — ■ '^iio — '''I'l.'o + '^yao — '^MO + " ('^lu-ij 21. > oH~ '^2i; 21- J j 0)
486. Theorem. The rotation groups, not dihedron groups, defiiie the
algebras given below :
(a) The tetraliedral group: generators Cj, e^ e? = 1 = cf = (e, Cg)'
Let u" = 1. The Fkobenius algebra is
'^IIO ^^-220 '*'.Tt(l ^UO '^-iSA ''-IfiO '''510 '"-650 '''SCO \iO ^50 ^va
^1 = '^IIO + '■■>" \'20 + <■■> "^^m + '^410 + '■■>" ''-650 + <■■> ?'CfiO
^2 = '^llO + ^^220 + '^SiliO J ('^^O + '^-IM + '^(Joo) + I (?-4Bo + /l,oo + ''-CW
+ ''-5«0 + ^MO H" ''■660)
(b) The octahedral group : e| = 1 = oil ^ (e^ e^)'-
Let w* = 1. The algebra is
'^UO ^^220 ^^330 ^^440 '''•340 ''-430 '^650 ^-fifiO ''-770 '^■SCO '^660 '^670
^^760 ^^070 '^760 '^880 ''■990 ''"aaO '^890 '''•980 '^8a0 ''-a«0 ''-9a0 '^a90
^1 = ''"IIO '^220 + '''•330 ''^440 + ''^550 + "' '^CBO + <J '^770 '■■^ ^m + ^^'^ '^990 + '■' '^aaO
e, = ;i„o + ^220— i ^330 + i^^- 3 X430 + M 1 + <^) ^sco + i (1 - w) ^5,0
+ i (1 - W) ^890 + i (1 + l.i) Xg^o + Jn^— 3 ;i3,o— i ;*-440
+ i (1 + O) ;inBO — ^ ^060 + ^ ^CTG + Ml — '■>) ^m + ^ " ?^980 + * ^.0
+ i (1 - <,,) X-,0 + h ;i:,-,o + i (,) a,:o + Ul + '^) A^^, + U<.90 - i <^ >.„ao
(c) The icosahedral group : ej = 1 = e| = (Cg Cj)-.
The algebra is X„o ^00 '^'.'o ?-,„;o ^sw where -
*,y= 2, 3, 4 7^, Z= 5, 6, 7 j>, <? = 8, 9, a, /? *, ^ = >-, .\ e, f, >?
487. Theorem. The group G^^^, e\=\=el = (e.c,)", defines the algebra
''"llO ^ijO ^^klQ '"ji./O ^slO ^uvG
where '
h3 = 2, 3, 4 Ic, 1= 5, 6, 7 ^j, y = 8, 9, a, ^, y, 8
s, t = e,^, r„ 6, I, X, X ti,v = ^, r, o, n, p, a, r, ^
488. Theorem. The groups defined by the relations e"= 1 =eS, e^' ^ e^=^e'^
m prime to a, give Fkobenius algebras of order r = ac which are sums of
quadrates as follows :
ajt-g of order 1 Itjkj of order (7;
'Shaw 6. «Fbobenids3; Dickson 4. »Poincare4.
128 SYNOPSIS OF LINEAR ASSOCIATIVr: ALGEBRA
where a^ is the highest cominon factor of m — 1 and a; g is, the lowest expo-
nent for which m'-' = 1 (mod a); c=- kg.
If a' is the smallest divisor of a, 4^ 1 , and az^a^ a', then m"' ^ 1 (mod a^).
If a" is the next smallest divisor of a, a = a^a", mP-=. 1 (mod a.^, and so
on for all divisors of a ; if also 4> {N) is the totient of {N), then
^{a) = hg ^ (a,) = /ii(7i •••• ^{aj)=^h9j {j = I, 2. . . . I— l,i -\- I . . . .p)
We notice that if gj is a primitive a-th root of unity, n a primitive c-th
root of unity
e, = l cA' '"' ?^^- » eo = 2 7t* (2 ^^j% \] o)
wherein
i = 1 hj, y = 1 g^ 1=1 7i^
The multiples of ap_^_,_i, namely v, a3,_:r+i, where Vt is prime to a^, are
divided into 7i^ sets of g^ each ; s^P is the lowest in the Z-th set, the set being
4"> ^4" • • • • »«''*~^4'') andy + 1 is reduced modulo^ g^..
489. Theorem. The algebra defined by the groups 1 = e^ = e^ = ej'
e.^ ej = gj 63 eg ej = fj e^ e., e^ = e^ 63 e^ is given by the forms "K occurring
in the equations
where
a; = p + l,^ 1,0 no = n r2j, + i=l 7^, i = 1 "p-x+i /=! «x
/_, is any integer < n^; and prime to n^. [has therefore ^ (%) values], /+ 1 is
reduced modulo n^; n^ is any divisor of w, the quotient^ being ??p_^ + i.
490. The papers of Frobenius and Burnside on group-characteristics
should be consulted.
•Shaw 13.
SPECIAL CLASSES OF GROUPS 1 29
XXVn. SPECIAL CLASSES OF GROUPS.
491. Since every group determines a Fuoi{f:Niu.s algebni, it is evident that
this algebra might be used to determine tlie group and to serve in applications
of the group. Since the group admits only of multiplication, the group
properties become those of certain numbers in the algebra combined only by
multiplication. However, if the group is a group of operators, or may be
viewed as a group of operators, it may happen that the result of operating
on a given operand may be additive, in which case the numbers of the algebra
become operators. Examples are given below.
492. Substitutions. Since every abstract group of order r is isomorphic
with one or more substitution groups on r letters or fewer, it follows that the
permutations or substitutions of such groups may be expressed by numbers of
the algebra corresponding to the abstract group. Thus a rational integral
algebraic function P of n variables may be reduced to the form
m
P=^ Pi
i=l
where Pj is expressible in the form
where Af is a positive or negative numerical coefficient and Sj is a substitution
of the symmetric group of the h variables. F^ is a rational integral algebraic
function of the variables. All the substitutional properties of P^ are direct
consequences of the form {A'p + + ^J;* yS'„). For example
-P = i «3 — i «a + 3af CTj — 3aj ag — i a| Og + I Oj a5 = Pi 4- Pg
where
P]= i [1 — (rto a-i) + («i Ws) — («1 «2 «3)] • «3
Pi= [3 — s («i a. Os) — 3 (rt, ttg) + I {(ii as)] . a; a,
wherein the bracket expresses an operation. We may find solutions for
equations such as
{l+a + a^ + a')P=0 <t = (abed)
or other forms in which the parenthesis is any rational integral expression in
terms of substitutions.
The solution of this particular case is P = (l — a) F, where F is any
rational integral function. These equations are useful in the study of
invariants.'
493. Linear Groups. A group of linear substitutions has corresponding to
it an abstract group, such that if the generating substitutions of the linear
group, H, are 2i, ^2 2p, with certain relations 2i, 2^, -r, = 1, 2,-, 2^
.... 2,., = 1, etc., then the abstract group is determined by generating sub-
stitutions (Ti, (Tj Op, with relations (7^,0^, a,, = 1, aj, cr^, <^r, = 1, etc.
'A. TocNO 1.
130 SYNOPSIS OF LINEAK ASSOCIATIVE ALGEBRA
If we choose a suitable polygon in a fundamental circle, the circle is
divisible into an infinity of triangles, which may be produced by inversions at
the corners of the polygon, according to the well-known methods. The
group G generated by (Tj, cTg. . . -cTp without the relations is in general infinite.
With the addition of the relations we get a group G' isomorphic with
E, H being merihedrically isomorphic with G.
Then G', or what is the same thing H, may be made isomorphic with a
Frobenitjs algebra, which is of use in the applications of the group. A notable
application of this kind was made by Poincare.^
This application is devoted on the one hand to the study of the linear
groups of the periods of the two kinds of integrals of a linear differential
equation of order n which is algebraically integrable; and, on the other, to the
proof that for every finite group contained in the general linear group of n
variables there is such a differential equation. The results are chiefly the
following :
494. Theorem. For every group G' there is a system of Fuchsian func-
tions, Abelian integrals of the first kind, such that if ^(2) is any such function,
and if S is any substitution of G' to which corresponds a linear substitution
on^, "^f, (a^-^X=l), then
where co is called a i^eriod of K(z).
There are also Abelian integrals of the second kind P{z, a), such that
P{zS, a) ■=. P(z, a) -\- ^{a), where ^(a) is the a-derivative of a function of the
first kind.
495. Theorem. The genus of the group being q, there are q independent
integrals of each kind ; all others are expressible linearly in terms of these.
496. Theorem. In the Frobenius algebra corresponding to the group let
X be 2 . -^iCi, where e,- corresponds to S^. Then KX means I.XiK{za~^), and
c)X means the period of KX corresponding to the period u of Kz. Then there
are three kinds of quadrates in the algebra.
I. Those for which KX^ = constant, for all values of K and any number
Xa in the a-th quadrate. In this case (dX= identically for any ^and any
substitution S, and if A'„ = 2X„;<'j, there are linear relations among the
coefficients X^i. Also P{z, a) X, is an algebraic function.
II. Those for which KX^ is constant for each K if X, is properly chosen,
BO that for any ^and any S there is an X^ such that uX^ = 0.
There is an integral K' whose periods are linear combinations with
integral coefficients of X^t ; this integral K' combined with K by Riemann's
' POINCAR^ 4.
SPECIAL CLASSES OF GROUPS 1 31
relations gives tlie coefficients of the periods o in uX = ^ JC^iUf-, that is,
determines uf^j. There are but q such relations independent.
Also P{z, a) X^ is not an algebraic function for this X^.
III. K . X^ is not constant for certain /(T's, and any X^. For any such
K we may write KX^ = (^(z), then the periods of G{z) being cji, cjg • • • • <-<,„
for the m substitutions of G\ if we forin the periods of G{za), we get the
same periods w in another order ; a group determinant may be formed from
these by letting a run through G, which must vanish as well as its minors of
the first in — n — 1 orders.
That there be a rational function of x, y, satisfying a linear equation of
order n, it is necessary and sufficient that there are numbers h^, h^ . . ■ ■ h^
whose group determinant is of the character above. There is thus always at
least one quadrate of the third kind.
497. Theorem. An integral of the first kind, K, helongs to a quadrate if
^X = constant, for any number X not in this quadrate, but KX is not con-
stant for all numbers in the quadrate.
An integral P{z, a) helongs to a quadrate if for all values of X not in this
quadrate P{z, a) X is an algebraic function ; but for some values of X in the
quadrate P{Zy a) X is not an algebraic function.
The number of integrals of the first kind belonging to a quadrate of
order a? is a multiple of a.
Any integral can be separated into integrals each of which belongs to a
single quadrate.
498. Theorem. The 2q periods of K{z) are subject to a linear transforma-
tion by each substitution S of G'. The totality of these linear transformations
furnish a linear group isomorphic with H.
The relations between the periods of P{z, a) are found by writing the
linear relations between K{z), K{zS^, K{zS.?i, etc., and differentiating them.
The derivatives are subject to linear transformations which also generate a
group isomorphic with H.
The second group is related to the totality of quadrates of the second
kind, the first group to the totality of quadrates of the third kind.
499. Modular Group. This has been studied by means of the commutative
algebras.^
500. Laurent' has made use of representations of linear substitutions by
quadrate numbers or tettarions, to derive several theorems. His processes are
briefly indicated below.
Theorem. If cr = 2 c^j \j, where
c„ = 1 ?■ = 1 11 Cij = —Cji i:^J
then the tettarion r = 2cr^ — 1 represents an oWAogrona? substitution, and the
' J. W. Young 1. 'Laukent 1, 2, 3, 4.
132 SYNOPSIS OF LINEAK ASSOCIATIVE ALGEBRA
orthogonal group consists of all such substitutions. In this case a represents
a sJceiv substitution.
501. Theorem. Every orthogonal substitution may be represented by the
product of tettarions of the type ^
w = /ln + ^22+ •••• +Kn + i^ii + hj) cos ^ + i^Hj — T^ji) sin <?>
Zji and 2,jj absent
502. Theorem. Tettarions of the type c and 1 + c (/l,^ + ^^i) produce
tettarions representing symmetric substitutions.
503. Theorem. Tettarions of the type 1 + /l,j produce tettarions which
represent substitutions with integral coeflRcients.
l....n
504. Theorem. If Tr=:2 a^^ Aj^ represents an orthogonal substitution, then
l....n
Tpj = 2 aip ttjg /ii^ gives a new group of linear substitutions. By similar
compounding of coefficients of known groups, new groups may be formed.
505. AuTONNE^ has applied the theory of matrices to derive theorems
relating to linear groups, real, orthogonal, hermitian, and hypohermitian. If
where Uij is the conjugate of the ordinary complex number a^j, then t is
symmetric if ? = t ; it is orthogonal if tt = 1 ; real if t=:t; unitary if
TT= 1 ; hermitian if t=t. In the latter case the hermitian form /. ^ (t) ^ >• 0.
[In this expression (r) acts on ^ as a linear vector operator]. If r is hermitian
there is one and only one hermitian ^ such that ^" = r, or <f = t*.
Theorem. That an n-ary group G can be rendered real and orthogonal
by a convenient choice of variables, the following conditions are necessary
and sufficient :
(1) G possesses two absolute invariants: a hermitian form I . ^(r)^ and
an n-ary quadratic form of determinant unity, P =■ I . ^(p)^.
(2) G having been rendered unitary by being put into the form t* Gt~^,
in the transform of P, p is unitary.
506. Theorem. Every tettarion is the product of a unitary tettarion by a
hermitian tettarion.
To put a into such a form we take t" = aa and d = af^; then a =: vt.
The literature of bilinear forms furnishes many investigations along
these lines.
I Cf. Taber C, 7, and other papers on matrices. ■ Autonne 1, 2, 3, 4.
MODULAR SYSTEMS 133
XXVm. DrPFERENTIAL EQUATIONS.
507. Pfajf's Equation. To the Bolution of the equation
Xj dxi + X^ dx., -f- . . . . -f- Xm dxm =
Grassmann' applied the methods of the Ausdahnungslehre.
508. La Place's Equation. This may be written y^y— -q. ft has been
treated by quaternionic methods in the case of three variables.^ Other equa-
tions and systems of equations which appear in physics have been handled in
analogous ways. The literature of quaternions and vector analysis should be
consulted.^ The full advantage of treating the general operator v ^s an
associative number, would .simplify many problems and suggest solutions for
cases not yet handled.
509. It is pointed out by Skill* that by means of matrices the operator
g2 02 32 ^ 82 3' 7 3'
^ = '' a.r^ + * 3/ + ^ dz"- + f dydz + ^ ~dzdx + ^ Tx 37
can be factored into
{«!, + (^-«i>) a^ + {9-n)^\ { s-^ + i' 37 + ? -3z }
p and q being matrices (tettarions).
Therefore any matrical function of x, y, z which vanishes under the
operation of either of these linear operators is a solution of the equation
A . 0=0
It is obvious that this method is capable of considerable extension.*
XXIX. MODULAR SYSTEMS.
510. It is obvious that every multiplication table may be expressed in
the form
gje^— 2yyfce*=0
If now we consider a domain admitting e,-, e^, etc., and their products and
linear combinations, it is evident that we have a modular system. The
expressions e^ . . . . need not be ordinary algebraic variables, of course ; they
may be function-signs, for example.
Every modular system may be considered to represent, and may be
represented by, an algebra. From this point of view all numbers are quali-
tative except integers.
>FoR8TTHE 1. Cf. Ausdehnungslehre, 1862, §§500-527.
' Boole 1; Carmichael 1, 2, 3, 4; Brill 2 ; Graves 1.
» WeDDERBCRN 3 ; POCKLINOTON 1. *BrILL3.
• Cf. B. Peikoe 2. Same In Appendix I In B. PeirceS.
9
134 SYNOPSIS OF LINEAR ASSOCIATIVA ALGEBRA
XXX. OPERATORS.
511. The use of different abstract algebras in forms which practically
make them operators on other entities is quite common in some directions.
In such applications the theory demands a consideration of the operands as
much as of the operators. As operators they have also certain invariant,
covariant, contravariant, etc. operands, so that the invariant theory becomes
important.
For example, the algebra of nonions plays a very important part in
quaternions as the theory of the linear vector operator.^
512. Invariants of Qnantics. The formulae and methods of quaternions
have been applied to the study of the invariants of the orthogonal trans-
formations of ternary and quaternary quantics.^ If ^ is a vector, then q^g~^
is an orthogonal transformation of ^, q being any quaternion of non-vanishing
tensor. Every vector or power of a vector or products of powers of vectors
furnishes a pseudo-invariant. Orthogonal ternary invariants are then those
functions of vectors which are mere scalars, the list being as follows :
r-p r-a Sa^ Spa^ Sa^y
In these, a, /?, etc. are practically different nablas operating on p, so that
we understand by S . a^y substantially what is also written aS.ViVsVs-
The formulae of quaternions become thus applicable to these symbolic
operators, yielding reductions, syzygies, etc. For example, the syzygies
Sa^ Syhe — S^y Sahe + Sl^h Says — S(5e So.yh =
Sap S^yh - Sl5p Sayh + Syp Sai^h - Shp Sa^y =
This amounts, of course, to a new interpretation of Aronhold's notation, and
the process may readily be generalized to n dimensions by introducing the
forms 7a/3, Ipp, lap, and the like.
513. Differential Operators. The differential operators occurring in con-
tinuous group-theory are associative, hence generate an associative algebra
(usually infinite in dimensions). Groups of such operators are groups in the
algebras they define, and their theory may be considered to be a chapter on
group-theory of infinite algebras. The whole subject of infinite algebras is
undeveloped. The iterative calculus, the calculus of functional equations,
and the calculus of linear operations are closely connected with the subject
of this memoir.^
■ See references under NonionB, previously given.
» McMahon 1 ; Shaw 14.
'PiNOHERLE 2, 3; Lemekat 1, 2, 3; Leac 1. The literature of this subject should be consulted.
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