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 INVARIANTIVE CHARACTERIZATIONS OF 
 
 LINEAR ALGEBRAS WITH THE 
 
 ASSOCIATIVE LAW NOT 
 
 ASSUMED 
 
 A DISSERTATION 
 
 Submitted to the Faculty op the Ogden Graduate School 
 OF Science I^f Candidacy for the Degree 
 of Doctor of Philosophy 
 
 department of mathematics 
 
 j 
 
 BY 
 
 CYRUS COLTON MACDUFFEE 
 
 Private Edition 
 
 Distributed By 
 
 THE UNIVERSITY OF CHICAGO LIBRARIES 
 
 CHICAGO, ILLINOIS 
 
 Reprinted from The Transactions of the American Mathematical Society, 
 Vol. 23, No. 2. March 1923 
 
Zbc Tantversiti? of Cbicago 
 
 INVARIANTIVE CHARACTERIZATIONS OF 
 
 LINEAR ALGEBRAS WITH THE 
 
 ASSOCIATIVE LAW NOT 
 
 ASSUMED 
 
 A DISSERTATION 
 
 Submitted to the Faculty of the Ogden Graduate School 
 OF Science in Candidacy for the Degree 
 OF Doctor of Philosophy 
 
 department op mathematics 
 
 BY 
 
 CYRUS COLTON MACDUFFEE 
 
 Private Edition 
 
 Distributed By 
 
 THE UNIVERSITY OF CHICAGO LIBRARIES 
 
 CHICAGO, ILLINOIS 
 
 Reprinted from The Transactions of the American Mathematical Society, 
 Vol. 23, No. 2. March 1923 
 
EXCK.---.fHI 
 
> »\ » » 
 
 INVARIANTIVE CHARACTERIZATIONS OF LINEAR ALGEBRAS 
 WITH THE ASSOCIATIVE LAW NOT ASSUMED* 
 
 BY 
 
 CYRUS COLTON MACDUFFEE 
 
 I. Introduction 
 
 We consider linear algebras in \vhich neither the commutative nor the asso- 
 ciative law of multiplication is assumed, and whose coordinates and constants 
 of multiplication are numbers of a general field F. A rational integral invariant, 
 or covariant, is a rational integral function of the constants of multiplication, 
 or of the constants of multiplication and the coordinates of the general number, 
 which has the invariantive property under the total group of linear homogeneous 
 transformations on the units. If an invariantive function also actually involves 
 the units, it has been called a vector covariant by Professor O. C. Hazlett.f 
 who shows that every rational integral vector covariant can be obtained as a 
 covariant of the general number of the algebra and a fundamental set of ordinary 
 covariants. 
 
 In Section II of this article, it is shown how vector covariants may be formed 
 directly from the constants of multiplication without assuming the knowledge 
 of any ordinary covariants or invariants. To do this, the notion is introduced 
 of a hypercomplex determinant whose elements obey neither the associative nor 
 the commutative law of multiplication, and a few simple properties of such hyper- 
 complex determinants are derived. From the vector covariants and the charac- 
 teristic determinants of the algebra, ordinary relative invariants may easily be 
 found. 
 
 In Section III the linear algebra in three units, one of which is a principle unit, 
 is considered. Invariants and covariants of the algebra are calculated by the 
 method of Section II, and a set of ten of these functions is shown to form a com- 
 plete system of invariants and covariants from the standpoint of Lie. 
 
 In Section IV it is shown that for the example of Section III the arithmetic 
 invariant denoting the rank can be replaced by a rational integral covariant. 
 The generic case is defined as the case for which certain three covariants are 
 
 * Presented to the Society, March 25, 1921. 
 t These Transactions, vol. 19 (1918), p. 408. 
 
 (135) 
 
136 C. C. MACDUFFEE [March 
 
 different from zero, and this case is reduced by rational transformations in the 
 general field to a canonical form. The parameters of the canonical form are 
 characterized by invariants. 
 
 II. Invariants and vector covariants 
 
 1. Notations. Consider the general linear algebra in n units whose multi- 
 plication table is given by 
 
 (1) eiCj = YlkZ" yuk^k {i, / = 1, . . ., m), 
 
 and whose general number is x = ^'il"xiei, where the yjj^ and Xj are num- 
 bers of a general field F. Consider the transformation of units 
 
 (2) T: e/ = J^fJl a^^ej Z? ^ | a,;,- 1 ?^ . (i = 1, . . . , «), 
 
 also with coefficients a,;,- in F. This transformation T induces upon the con- 
 stants 7y-4 of multiplication and the coordinates Xi a transformation 5 which 
 carries 7,;,^ into yijk and Xi into x, in such a way that 
 
 (3) eWj = ^IZ" y'ijke'k, 
 and 
 
 (4) E:"% = ^ = ^'=Z':i^'^^ 
 
 According to Professor Hazlett, * a vector covariant is defined to be a function 
 of the units and the constants of multiplication and the coordinates of the form 
 
 'vilijk; x/yBs) (i, j,k,r,s = \, ..., n), 
 
 such that, under a transformation T of the units and its induced transformation 
 S, there exists a relation of the form 
 
 (5) v' = v{y'ijk; xl; e's) = D" v{y,jk; x/, e^ {i, j, k, r, s = 1, . . ., n). 
 
 The integer n is called the weight of the vector covariant v. 
 
 As Miss Hazlett noted, f it follows readily from (1) and (3) that every vector 
 covariant is expressible linearly in the units, i. e., 
 
 (6) ^' = T^:^ie:. v = Y:rjiViei, 
 
 where v'i is the same function of the y'ij^ and x'^ that f, is of the 7,;,^ and the x,. 
 
 * These Transactions, vol. 19 (1918), p. 408. 
 t Ibid., p. €16. 
 
1922] UNBAR ALGEBRAS 137 
 
 2. Theorem 1. The coefficients Vj of every vector covariant of weight n expressed 
 in the form v = SjC" ^,-^,- are transformed cogrediently apart from the factor D~'' 
 mith the coordinates Xi of the general number x = 2)~"xj^,- under a linear trans- 
 formation of determinant D. 
 
 From (5) and (6) we have 
 
 Making use of (2), we find that 
 
 On account of the linear independence of the units Cj, which occur only where 
 shown explicitly, 
 
 (7) v;= D-" 2];:>.' «.>• • {j = !,:.., ft). 
 
 Similarly from (4) and (2) we have 
 
 (8) Xj = YPlxla^j (/=!,...,«). 
 
 Comparing (7) with (8), we see that the theorem is proved. 
 
 3. Hypercomplex determinants.* On account of the fact that multiplication 
 is usually neither commutative nor associative, a determinant whose elements 
 are hypercomplex numbers must be defined more precisely than a determinant 
 whose elements are ordinary numbers. We define the general hypercomplex 
 determinant of the nth order 
 
 (9) • D = 
 
 an ... ai„ 
 
 ani       a„„ 
 [oiia22 • . . a„„] 
 
 to be the sum of n ! terms of the type 
 
 (-1)* [ai,-, 02,, ••• a„ij 
 
 in which ii, 4 i„is an arrangement of 1, 2, . . .,n derived from the natural 
 
 order by k interchanges. The first subscripts must occur in their natural order 
 in every term. The brackets indicate that the method of grouping the factors 
 is arbitrary, but the same method is to be used in each term. A particular 
 
 * Determinants whose elements are quaternions and hence associative but not commuta- 
 tive were considered by Cayley, On certain results relating to quaternions, The Philosoph- 
 icalMagazine, vol. 26 (1845) pp. 141-145; and also by C. J. Joly, second edition of 
 Hamilton's Elements of Quaternions, vol. 2, Appendix 1, p. 361. 
 
138 
 
 C. C. MACDtnPFEE 
 
 [March 
 
 hypercomplex determinant of the nth order is obtained by replacing the brackets 
 inclosing the leading term by a particular grouping of that term, in which case 
 it is understood that every term of the expansion is to be grouped in that way. 
 Thus there are as many particular hypercomplex determinants of the nth order 
 as there are ways of grouping n ordered factors. 
 
 4. Lemma 1. Every hypercomplex determinant merely changes sign upon the 
 interchange of any two columns. 
 
 Consider the determinant (9). Its terms may be arranged in gC^!) pairs 
 of the type 
 
 (10) 
 
 (-1)* [axi^ ... Or, 
 
 (-1)*+' [au. 
 
 . . Qri. a,, 
 
 
 where the two terms in each pair differ only in having their i, and i^ subscripts 
 interchanged. Since each is obtainable from the other by one interchange of 
 subscripts, they are opposite in sign. Let us denote D with its rth and 5th 
 columns interchanged by D'. Evidently this interchange leaves the first sub- 
 scripts in their natural order but interchanges the second subscripts v and i, 
 wherever they occur, and so interchanges the absolute values of the terms in 
 each pair (10). Thus every term in D with its sign changed is equal to a term 
 of D' and vice versa. Then D' = —D. This argument depends in no way 
 upon the manner of grouping, so the lemma holds for all hypercomplex deter- 
 minants. 
 
 No analogous theorem exists concerning the interchange of two rows. 
 
 Lemma 2. Any hypercomplex determinant two of whose columns are identical 
 is zero. 
 
 For by the interchange of the two identical columns the determinant is un- 
 altered, and yet changes in sign. 
 
 5. Lemma 3. A hypercomplex determinant the elements of whose jth column are 
 binomials a^j -\- bij is equal to the sum of two determinants identical with the first 
 except that the jth column of the one is composed of the a,j- while the jth column of the 
 other is composed of the b^j. The corresponding theorem holds for rows. 
 
 Let us set 
 
 D = 
 
 an • • • ihr + Clr)       flln 
 
 a»l • • • {bnr + O • • • flnx 
 
 A = 
 
 ail       i>lr     • (^In 
 
 Obi • • • b„r . .   a„„ 
 
 A = 
 
 On ... Clr ... Oin 
 
 flnl • • • C„r . . . a„ 
 
1922] 
 
 LINBAR ALGSBRAS 
 
 139 
 
 where it is understood that the manner of grouping is arbitrary, but the same 
 manner must be used consistently in every term of each determinant. It 
 follows from the distributive law that 
 
 (11) a„- 
 
 (Psi^ + Csi) . . . a„i^ = au^ . . . bsi^ 
 
 + au ... Cs, 
 
 for all values of the subscripts and for all methods of grouping. It holds in 
 particular for i^ = r and 5 = 1, . . ., n where r is fixed. Let k denote the num- 
 ber of interchanges necessary to obtain the order ij, . . . , i„ from the natural 
 order 1, . . ., n. Then, multiplying (11) by ( — 1) and summing for ii, . . ., i„, 
 we have 
 
 D = Di + Di, 
 
 which was to be proved. 
 
 We prove the corresponding theorem for rows by noting that (11) holds when 
 5 is fixed and i^ ranges over the values 1, . . .,n, and summing as before. 
 
 6. Theorem 2. The determinant 
 
 V = 
 
 ei ... e„ 
 
 ei 
 
 is a vector covariant of weight 1 of the linear algebra in n units ei, 
 manner of grouping. 
 We set 
 
 V' = 
 
 Under transformation (2) this becomes 
 
 . , e„ for every 
 
 V 
 
 ^:;:>i,., ...^;::>„,,.., 
 
 Z';:r«iu^^....Z'::r««^n^.. 
 
 From Lemma 3 we see that 
 
 V, = 
 
 <i »,=i 
 
 "i,-, e,^ 
 
 «n.-„ ei. 
 
 "ifi «.i 
 
 By Lemma 2 such of these determinants as have two identical columns are zero. 
 Moreover the as are numbers of the field F. Hence we may restrict the I'l, 
 . . . , t„ to sets of n distinct values and write 
 
140 
 
 C. C. MACDtTFFBE 
 
 [March 
 
 F' = !:«:,.. 
 
 • ei„ 
 
 e„ ... e,. 
 
 Let k denote the number of interchanges necessary to produce the order j'l, 
 . . .,i„ from the natural order of the integers 1, . . .,n. By Lemma 1 the above 
 determinant changes sign with each such interchange, that is with each inter- 
 change of two columns. Then 
 
 €t ... € 
 
 V'^J2i-'^)'au,... a,.-. . ".' ." 
 '■ ■'■" e,...e„ 
 
 The above sum is exactly the determinant D of the transformation (2). Hence, 
 as stated, 
 
 V = DV. 
 
 7. Rational integral invariants. In the general linear algebra in n units the 
 right- and left-hand characteristic determinants* 
 
 5'('^) = I'^ZiyiJk^j - ^ik<^\ (i, j.k = 1, ..., n), 
 
 (12) 
 
 and the coefficients of the powers of co therein are absolute covariants under the 
 general group of homogeneous linear transformations on the units. In view of 
 Theorem 2, there are conceivably as many vector covariants of weight 1 as there 
 are ways of grouping n ordered factors, but it may happen in certain algebras 
 that a smaller number results. Thus for commutative algebras they all vanish 
 and for associative algebras they become identical. By means of the multi- 
 plication table (1) each such vector covariant can be expressed linearly in terms 
 of the units, i. e., in the form (6). Suppose that p of them are linearly indepen- 
 dent. By Theorem 1 the coefficients Vi are transformed cogrediently apart from 
 the factor D~^ with the coordinates a;, of the general number. Therefore if 
 these coefficients ^,- be inserted in place of the Xj in a homogeneous absolute 
 invariant of degree r, there results a relative invariant of the linear algebra of 
 weight r. This method yields 2np relative invariants. 
 
 III. A Lie complete system of invariants and covariants for 
 
 A TERNARY ALGEBRA 
 
 8. Foreword. It will be shown that for the algebra with three units, one of 
 which is a principal unit, the invariants formed by the method of the preceding 
 section together with those obtained from the characteristic determinants by 
 
 * h. E. Dickson, Linear Algebras, Cambridge Tract No. 16 (Cambridge, 1914), p. 17 and 
 p. 20. 
 
1922] LINEAR ALGEBRAS 141 
 
 the methods of the theory of invariants of forms constitute a complete system 
 of invariants and covariants from the standpoint of Lie. In fact, if all the 
 invariants obtainable by both methods are taken, there is considerable redun- 
 dance. A set of the simpler ones is shown to be composed of functionally inde- 
 pendent invariants. 
 
 9. The finite transformations. In the linear algebra in three units 1, d, e^, 
 where 1 is a principal unit, the general number is of the form 
 
 (13) X = Xo -\- Xiei + X2^2 
 
 where xo, Xi, xi are numbers of a field F. The multiplication table may be taken 
 to be 
 
 <?i^ = ao + ai^2 + 0262, 
 Q .X e>^ = bo + biei + 62^2, 
 
 ^1^2 = Co + Ciei + €202, 
 
 eiCi = c?o + die\ -f- d2e2, 
 
 where the coefficients are numbers of F. By applying the linear transformation 
 
 of units • 
 
 /, c-v ^1 = ao + ociBi -\- a2^2, 
 
 ^ ' €2 = Po + /3iei + ft^2, 
 
 where the coefficients are numbers of F such that D ^ 01182 — «2/3i is different 
 from zero, the principal unit is left invariant and the general number and multi- 
 plication table become (13) and (14) respectively with each letter primed. That 
 is, the transformation (15) of units induces upon the coefficients the following 
 transformation : 
 
 x'a = Xo + D-\a2&o - oio^2)xi + D-^(aopi — a;i/3o)^2, 
 x[ = D-i ^a-Ti - £>-» fiiX2, 
 X2 = — D~^ UiXi + D~^ aiXi, 
 
 a'o= A+ D--^{a2po - "oft)^ -f- D-^aoft - ai/3o)C, 
 a[ = Z?-i P2B - D-^fiiC, 
 a'2 = -D-i ajS -f- Z)-i aiC, 
 (16) ^'o = -E + D-^(a20o - ao02)F + r>-i(aoft - a,0o)G, 
 
 b[ = Z?-i ftF - D-i ^iG, 
 b!, = - Z?-i a2F + Z?-» aiG, 
 
 c'„ = H + D-^{a20o - aoffi)! + D-'{ao0i - a,0o)J, 
 c[ = D-^ 02l - -D-i 0ij, 
 C2 = - D-^ a2l + D-' aj, 
 
 d'o = K + D-\a2^o - ao02)L + Z?-i(aoft - ai(3o)M, 
 d[ = D-» ^L - D-i /3iM, 
 d'2 = - D-^a2L + D-'aiM, 
 
142 C. C. MACDUFFEE [March 
 
 where 
 
 A = ao^ + ai^'ao + 02^60 + cL\aiPa + aias^o, 
 
 B ^ a-^a\ + a-^h\ + 2aoai + a\aif:\ + ctiai^x, 
 
 C ^ 01.^0.2 + oi^hi + 2aoa2 + aia2C2 + «i«2rf2> 
 
 £ = ^0' + /Si^ao + fii'ba + /Si^s^o + &iM<>, 
 
 F = |3i=ai + ^2^61 + 2/3o|3i + /Siftci + /SiMi, 
 
 6^ = /3i%2 + ^^h + 2/3o;32 + PACi + fiyl3^2, 
 
 H = ao/3o + ai/SiCo + 02/3260 + aiftco + a2/3ido, 
 
 7 = ai/3o + ao/3i + ai/SiOi + 02/3261 + ai/32Ci + a20idi, 
 
 J = a2/3o + ao/32 + ai/3ia2 + 02/3262 + ai/32C2 + a^ffich, 
 
 K = Qo/3o + oi/Jiao + 02/8260 + a2/3iCo + ai^^o, 
 
 L = ao/3i + oi/3o + oijSiOi + 02/3261 + 02/3iCi + oi/32(ii, 
 
 M = 00P2 + 02180 + ai/3ia2 + 02/8262 + 02/3iC2 + ai^zdi. 
 
 10. The Lie group. The generators* of the infinitesimal transformation 
 corresponding to the finite transformation (16) are found to consist of six partial 
 differential equations which it is not necessary to give here in detail. The 
 first term of each equation involves only the Xi, and is the only term in which 
 the Xi occur. Since these equations are generators of an infinitesimal group 
 corresponding to the finite group of transformations (16), they form a fortiori 
 a complete system of partial differential equations. The six equations in fifteen 
 variables have nine functionally independent solutions, and these solutions form 
 a complete set of absolute invariants and covariants of the linear algebra (14). 
 
 Moreover, if the terms involving Xo, Xi, X2 in these equations be deleted, there 
 results another complete system of partial differential equations, the generators 
 of the group corresponding to the group of transformations in (16) on the con- 
 stants of multiplication only. This system has six functionally independent 
 solutions, the six absolute invariants of the linear algebra. They are six of the 
 nine solutions of the first complete system. We see then that there are exactly 
 six absolute invariants and three absolute covariants in a complete system of 
 invariants and covariants for the algebra. There are then no more than seven 
 functionally independent relative invariants. A complete system will be 
 exhibited of three absolute covariants and seven relative invariants whose 
 jacobian does not vanish identically. 
 
 11. Invariants of the characteristic determinants. The right- and left-hand 
 characteristic determinants (12) become 
 
 5(co) = - o)' -h Aoj^ - Tco -f- A, 
 5'(a)) = - 0)5 -f- A'co" - r'w -f- A', 
 
 • Lie-Scheffers, Vorlesungen iiber continuierliche Gruppen, Leipzig, 1893, p. 716, et seq. 
 
1922] LINEAR ALGEBRAS 143 
 
 where 
 
 A = Sjco + (ai + ct)xi + {di + b^X2, 
 
 r = Zxi? + {ayCi — a2Ci — ao)xi^ + (Wi — hdi — 60)^2^ + (2ai 
 + 2c^Xiix-i. + (262 + 2di)xtiXi + (0162 — 02^1 + c-4i — Cid2 —do — Co)xiXt, 
 
 (17) A = a;o' + (02^0 — aoC2)xi' + (6i(io — Mi)«2' + (ciiCi — OjCi 
 
 — Co) xoxr^ + (ai + C2)xo^Xi + (62 + c/i)x:o^X2 + (Wi — hd^ — 60) xoXi^ 
 + (coC?2 — Cido + 02^0 — 00^2 + OoCi — aiCo)xi''x2 + (60^2 — Wo + Qo^i 
 
 — aibo + CiC?o — Codi)xix^'^ + {c^di — cA + aib2 — 0261 — co 
 
 — do)^0^1^2> 
 
 and A', V, A' are obtained from A, T, A respectively by interchanging c, and 
 d,-, for i = 0, 1, 2. Each of these six expressions satisfies the equations of §10 
 and is an absolute covariant of the algebra. 
 
 The invariants of the six ternary forms (17) are invariants of the algebra. 
 A and A' have no invariants. F and T' have one invariant each, their hessians. 
 The ternary cubic forms A and A' each have two relative invariants, the 5 and 
 T of Aronhold.* But only six of the seven invariants have been accounted for 
 by this means, even if all six should prove to be independent. Thus it is evident 
 that the invariants obtainable as the invariants of the coefficients of the char- 
 acteristic determinants are not sufficient to form a complete system. 
 
 12. Additional invariants by the method of Section II. By Theorem 2 of §6, 
 the hjrpercomplex determinant 
 
 1 
 
 ei 
 
 €2 
 
 1 
 
 e\ 
 
 62 
 
 1 
 
 ei 
 
 e2 
 
 V ^ 1 e\ €2 = ^1^2 ~ B2P1 
 1 ei €2 
 
 is a vector covariant of weight 1. By the multiplication table (14) this can be 
 expressed as 
 
 (18) F = (co - rfo) + (ci - d,)ei + (C2 - ^2)^2- 
 
 Now by Theorem 1, §2, the coefficients of the units in (18) are transformed co- 
 grediently apart from the factor D~^ with the coefficients Xo, Xi, X2 of the general 
 number, and hence when substituted for these variables in (17) give relative 
 invariants of the algebra. f Thus we have 
 
 Ai = 3(co — do) + (ai + C2)(ci — di) + (di + 62) (^2 — £^2), 
 
 A[ = 3(co - do) + (fli + d2)(,ci — di) + (ci + 62) (C2 - £^2), 
 
 r2 = 3(co — do)^ + (aiC2 — a2Pi — ao)(ci — diY + •   • , 
 
 Tj = Z{co -^ doY + (aA — a^i — ao){c\ — diY +   • • , 
 
 A3 = (co — doY + (02^0 — aoC2)(ci — diY +       , 
 
 A3 = (co — doY + (.Oido — aod2){ci — diY + • • • . 
 
 (19) 
 
 * Salmon, Higher Plane Curves, Dublin, 1897, pp. 191-192. 
 
 t Throughout this article the subscript on the symbol for an invariant indicates its weight. 
 
144 C. C. MACDUFFEE [March 
 
 Each of these forms is transformed into a function of itself by the differential 
 operators of §10. 
 
 Evidently there are functional relations between the twelve relative invariants 
 which these methods yield, since it has been shown that there can be but seven 
 functionally independent invariants. This redundance makes unnecessary 
 the use of the complicated 5 and T invariants of A and A'. The 5 invariant of 
 A — A' is quite simple however, and will be used. 
 
 13. A complete system of invariants. If will be shown that the following 
 three covariants and seven invariants form a complete system : 
 
 (20) A, A', e = r - r', H^ = hessian of V, H^' = hessian of T', 
 
 54 = 5 of A - A', Ai, Tj, r/, A3. 
 
 Since we have only ten relations in -fifteen variables, it is sufficient to show 
 that they are independent when five of these variables are put equal to constants. 
 It is found convenient to set ao = ai = C2 = t/s = and a2 = I. In fact this 
 normalization can be made upon T, V, A, A' before H2, H2', and Si are calculated. 
 The ten invariants (20) then reduce to the fairly simple forms: 
 
 A = 3:1^0 + (^2 + di)xi, 
 
 A' = Zxa + (62 + Ci)x2, 
 
 9 = — (ci — di)xi^ — biici — ^1)^2' — 2(ci — di)xQX2, 
 
 H2 = 4ci{di^ - Ml + 62^) + Uboci - 3bi^ - 66i(co + d,) 
 
 -3(co + do)^ 
 
 Hi' = 4di(ci^ - biCi + 62') + 12bodi - 3bi^ - 66i(co + do) 
 
 (21) - 3 (Co + d^y, 
 
 Si = (ci - diY/Sl [3bi{co - doY + 6{ca - d„){c,d^ - c^dO 
 
 + 3b, (Co - do){ci - di) - {ci - diy (V + 3bo)l 
 Ai = 3(co — do), 
 T2 = 3(co - doY - c-i(ci - dO\ 
 T2' = 3{co - do)' - di(ci - d,Y, 
 A3 = (co — do)^ + co(ci — di)^ — Ci(co — do){ci — di)'^, 
 
 14. Independence of the invariants. To prove the independence of these 
 ten invariants it is sufficient to prove that the jacobian 
 
 (22) d(A, A', e, H2, Hi', Si, Ai, T^, T^', A3) 
 
 i>{xo, xi, Xi, bo, bi, bi, Co, Ci, do, d\) 
 
 does not vanish identically. Now only the first three of these polynomials (21) 
 involve xo, Xi, X2, and only the first six involve bo, bi, bi. Hence it follows from 
 considering the Laplace development that the jacobian (22) factors into three 
 factors, viz., 
 
1922] LINEAR ALGEBRAS 145 
 
 d(A, A^ e) ^ bjHj, Hi', Si) ^ a(Ai, Fa, ^2^ A,) 
 
 b{xo, xi, X2) d(6o, 61, 62) d(co, ci, rfo, ^1) 
 
 It is sufficient to show that no one of these three jacobians vanishes identically 
 in ci, and hence it is sufficient to show that the coefficient of the highest power of 
 Ci in each jacobian is not identically zero. It is then permissible to drop all 
 terms in each element except those involving Ci to the highest power to which 
 it occurs in that element; for the other terms evidently cannot enter into the 
 term of highest degree in Ci in the expansion of the determinant. It is important 
 to be sure, however, that the coefficient of the highest power does not vanish, 
 for this method does not give the coefficients of lower powers correctly. By 
 this method it is easy to show that 
 
 d(A, A', e) „ , , , , 
   ' = oxict' + lower powers of Ci, 
 
 d(Xn. X^. X-)) 
 
 b{Xo, Xu Xi) 
 
 oiHi, Hi , Si) 16 , , I \ s 1 1 r 
 
 = Oi(co — da) Ci^ + lower powers of Ci, 
 
 i>(bo,b„bi) 9 
 
 d(Ai, r^, Ti', A3) ^ _ g^^, ^ j^^^^ p^^^^^ ^j ^^_ 
 
 d(co, ci, do, di)   
 
 Then the jacobian (22) becomes 
 
 96 Xidi{co — do)ci'^* + lower powers of Ci, 
 
 and the ten polynomials (20) are functionally independent and form a complete 
 system of invariants and covariants of the linear algebra (14) from the stand - 
 point of Lie. 
 
 IV. Characterization by invariants of a canonical form 
 
 15. The rank covariant. Let us consider the algebra in three units 1, d, et 
 whose general number and multiplication table are given by (13) and (14) 
 respectively. The rank* of every such algebra is three or two according as every 
 number does not or does satisfy a quadratic equation. This rank is an arith- 
 metic invariant under every linear transformation of units (15). It will be shown 
 that for this example the arithmetic rank invariant can be replaced by a rational 
 covariant. 
 
 * There is an equation p(oi) = of lowest degree having X as a right-hand (left-hand) 
 root. The degree of this equation is called the right-hand (left-hand) rank of the algebra. 
 (Cf. Dickson, Linear Algebras, p. 23.) It can be shown that there is at least one number 
 satisfying no equation of lower degree. 
 
146 C. C. MACDUFFEE [March 
 
 In every algebra of rank 3 with a principal unit, there is some number x 
 which does not satisfy a quadratic equation. Then the powers 1, x, x"^ are 
 linearly independent and may be taken as the units 1, ei, e^. That is, the multi- 
 plication table of every rank 3 algebra in these units has the form (14) with 
 Oo = ai = 0, 02 = 1. Conversely, every algebra of this form is of rank 3. 
 Evidently, then, a necessary and sufficient condition that an algebra (14) be of 
 rank 3 is that it be possible to make the above normalization. 
 
 From equations (16) it is seen that the conditions on a transformation (15) 
 which shall make ao' = a/ = and 02' = 1 are 
 
 A + D-\a20o - ao02)B + D-'iao0, - a^C = 0, 
 D-'ftS - D-ij3iC = 0, 
 .-D-^UiB + D-^aiC = 1. 
 
 These conditions are readily found to be equivalent to 
 
 (23) /3o = A,   ^1 = B, /32 = C, 
 
 where A, B, C, given in (16), are polynomials in the as and the constants of 
 multiplication. Thus the as can be chosen arbitrarily and the P's calculated 
 from the above relations, provided only that the determinant of the transforma- 
 tion 
 
 (24) D = Oztti' + (— fli + C2 -f 62)01^02 + (62 — Ci — di)aiai^ — bia^^ 
 
 be different from zero. Now a\ and a^ can evidently be chosen so that D is 
 not zero unless every coefficient of D vanishes. Then a necessary and sufficient 
 condition that the algebra be of rank 3 is that not every coefficient 02 etc. of 
 (24) vanish. 
 
 It will now be shown that there exists a covariant $ whose coefficients are the 
 coefficients of (24). From (17) we form the absolute covariant 
 
 r' - r + I (A2 - A'2) 
 
 every term of which involves (ci — <fi) or {d — d^). It was shown in (18) that 
 (co — da), (ci — dC}, (c2 — d'i) are transformed cogrediently apart from the 
 factor D~^ with xq, xi, Xi, so we may substitute X\ and xi respectively for these 
 expressions and obtain a relative invariant of weight — 1, 
 
 $ = a^-^ + (— fli + C2 + d2)x^X2 -\- (62 — Ci — di)xxx^ — h\X^, 
 
 whose coefficients are the coefficients of (24). Hence a necessary and sufficient 
 condition that an algebra (14) be of rank 3 is that $ does not vanish identically. 
 
1922] LINEAR ALGEBRAS 147 
 
 16. Determination of a canonical form. We shall designate as the generic 
 case that case for which neither $, r2 — Tj', nor A — A' vanishes identically, 
 and reduce this generic case by rational transformations in the general field F 
 to a canonical form — i. e., to a form in which each remaining coefficient is a 
 relative invariant under the most general transformation which does not destroy 
 the normalization. Since $ is not identically zero, the multiplication table can 
 at once be reduced as in §15. Then from (23) we see that in order to preserve 
 this normalization the ^'s must obey the conditions: 
 
 ^0 = "0^ + ai^ba + aia^ica + d^), 
 
 (25) /3i = ai'bi + 2aoai + aia^{ci + d,), 
 
 /32 = ai^ + 0:2^62 + 2aoa2 + a\ai{ci + ^2). 
 
 If C2 — di is not already zero, it is possible to effect a transformation making 
 Ci — di equal to zero. From (16) it is seen that under every transformation, 
 
 (26) C2' - J2' = - aiid - di) + ai(c2 - di). 
 
 Therefore if we choose ai and 0:2 satisfying the conditions 
 
 /g^N «i('^2 ~ ^2) = a^ici — di), 
 
 D = ai' + (c2 + di)ai^a2 + (62 — Ci — di)aia2^ — bia2^ 7^ 0, 
 
 and the /3's according to (25), we have the required transformation. Multiply- 
 ing the second equation of (27) by (c2 — ^2)' and eliminating ax by means of the 
 first equation, we have 
 
 D = ai'[(Cl - d,y + (C2 + d2){C2 - d^)(cx - d,Y 
 
 + (62 - 61 - Ci - d,){c2 - d,Y]. 
 
 Hence this normalization is possible if and only if the expression in brackets is 
 different from zero. But this expression is precisely the reduced form of the 
 invariant r2' — r2 which we have assumed not zero. 
 Since the covariant 
 
 A — A' = (c2 — dijxi — (ci — dijXi 
 
 does not vanish identically for this case, we know that Ci — di is not zero. From 
 (26) it is then seen that a necessary and sufficient condition on a transformation 
 that shall leave this normalization undisturbed is that a2 = 0. Then condi- 
 tions (25) require that ai 7^ 0,_ 0:2 = 0, j3o = ao^ .81 = 2aoai, ft = ai^, and 
 hence D = ai'. We see from (16) that by (15) 
 
 (28) C2' = 3aoai -f- ai^i. 
 
148 C. C. MACDUFFEE [March 
 
 SO let US now apply the transformation 
 
 /ao ai ajN / —Ci/3 1 0\ 
 
 Uo iSi ft/ " V C2V9 -2c2/3 1/ 
 
 which reduces C2' to zero. Dropping accents, we obtain the canonical form of 
 the generic case for rational transformations, viz., 
 
 ei^ 
 
 = 
 
 
 
 e2, 
 
 e-? 
 
 = 
 
 bo 
 
 + biei + 62^2, 
 
 exBi 
 
 = 
 
 Co 
 
 + CiCi, 
 
 
 e^ei 
 
 = 
 
 do 
 
 + diei, 
 
 
 (29) 
 
 ci - di ^ 0. 
 
 It is evident from (28) that for C2 = and ai 9^ 0, a necessary and sufficient 
 condition that C2' = is that ao = 0. Then the most general transformation 
 which will leave (29) unaltered in form is of the type 
 
 /O ai \ 
 
 (0 J'^ = 
 
 The effect of this transformation on the coefficients of (29) is to make 
 
 (30) bo' = ai%, bi' = ai^bu b^ = ai^bi, Co = ai'co, c/ = aiVi, 
 
 do' = aiHa, d\ = ai^d\. 
 
 Hence no further reduction by rational transformations is possible, although 
 by a transformation in general irrational any one of these coefficients which is 
 not zero can be reduced to unity. Then the other six are parameters which 
 cannot be altered except by a factor which is a root pf unity. 
 
 17. Characterization of parameters. It is seen from (30) that the most that 
 any transformation can do to (29), preserving its form, is to multiply each co- 
 efficient by a power of the determinant. It is natural to expect then that each 
 coefficient in (29) is the reduced form of a relative invariant of the original form 
 (14). We shall expect co, do and bi to be relative invariants of weight 1 ; Ci, di 
 and 62 of weight | ; and 60 of weight |. We prove that this is so by actually 
 finding these invariants in terms of known invariants. The invariants which 
 characterize co, ci, do, di will be denoted by, respectively, C, C, D", D'. 
 
 Making the normalization indicated in (29), we find that the invariants (19) 
 reduce to the forms 
 
 Ai = Ai' = 3(co — do), 
 
 ,„.. T2 = 3(co - ^0)== - ci(c, - d,y, 
 
 ^^^^ . r^' = 3(<:„ - do)' - d,{c, - di)\ 
 
 A3 = A3' = (co — do)^ + ca{ci — di)' - Ci{cq - do){ci - di)^. 
 
As - 
 
 -^A,3 + |A:(|A,= 
 
 - r^) 
 
 A3 - 
 
 
 -r,') 
 
 1922] LINEAR ALGEBRAS 149 
 
 We have exactly four relations to determine four unknowns. We find Co in 
 terms of the reduced invariants (31) and define C to be the corresponding func- 
 tion of the complete invariants, so that C is an invariant which reduces to 
 Co under the normalization (29) . Similarly we find invariants which reduce to 
 the other parameters. Thus, since Ti' — r2 5^ 0, 
 
 .C 
 
 (r^'-r,)" (r^^ - r,)^ 
 
 Evidently the weights are as predicted. 
 
 18. Equations defining the remaining parameters. Let us now consider the 
 problem of finding invariants which characterize ^o, bi, b^. In addition to the 
 invariants (31) we use the hessians H2 and H^' of T and r', and the invariants 
 54 and Te of A — A'. Under the normalization (29) we find 
 
 ia)H,, {b)H,', (c) 54 as given by (21), 
 ^ ^ (d) Ti = 4l^mp^ - ZlVp^ + QpHqrs - 12 mrpq^ + 8gV' - 27 p\h^, 
 
 where 
 
 I = Ca — do, m = bo(ci — di) — 6i(co — do), P = ^{ci — di), 
 q = ^(ci - di), s = J [62(^0 - ^0) + 2cido — 2codi], 
 r = —^bi(ci — di). 
 
 Making use of the relation (33) in (32c), we have (34) : 
 
 (33) F2 -H2' = 4(ci -Ji) (622 + 3^0 -ccfi); 
 
 (34) 61 (co - doKci - d,) + b,{co - d,Y = i [81 54/(ci - d,) 
 
 - 6(co - do) (ciJo - c^i) +\{ci- d,){Hi - H^') + Cidi{ci - d,)^. 
 
 Multiplying (326) by Ci and subtracting this product from the product of (326) 
 by di, we obtain 
 
 (35) 36i2 + 66i(co + do) + ^b^idx = {d^H^ - CiHi')/{ci - d,) 
 
 + Ac,d, (ci + dx) - 3(co + do)\ 
 
130 C. C. MACDUFFEE 
 
 Directly from (33) comes 
 
 (36) h' + 3bo = ^^^^ _ ^^^^ IH2 - H2' + 4cMci - di)]. 
 
 Now Co, do, Ci, di are the values which the known invariants C, D", C, D' 
 take when the general algebra becomes (29). Consider functions B", B', B' 
 of the coefficients of the general algebra such that B" has the value bo, B' the 
 value bi and B" the value 62 when the general algebra becomes (29). Then 
 corresponding to (34), (35), (36), and (32d), we have 
 
 (37) 
 
 where 
 
 5/(^0 _ J50)((;7' _ D') + 5"(C'' - D") = E, 
 
 35'2 + 65'(C« + £>") + 4B"C'D' = F, 
 
 B"2 + SB" = G, 
 
 76 = 4 Z^wps - 3 Pr^pi + 6 ^2;^^^ _ 12 nirpq^ + 8 gV' - 27 ^2g2^^ 
 
 £ = i [81 S4/(C' - P')' - 6(C'' - DO)iC'Do - CD') 
 
 + \ iC' - b'){H^ - H,') + C'D'iC - D')^], 
 F = (ZJ'Fa - C'H2')/{C' - D') + AC'D'{C' + D') - 3(C'' + £"')^ 
 
 ^ = (4C^ - 4P0 ^-^^ ~ ^' + 4^'-^'(^' - ^')1' 
 Z = C - Z?», w = Bo{C' - D') - B'iC - Z?"), 
 
 • f = - i (C - D'), q = - I (C - D'), 
 
 ;• = _ I B"(,C' - D'), s = I [S'XC - £">) + 2C'D'> - 2C»£)']. 
 
 Equations (37) determine the invariants 5", J5', B" uniquely. A practical 
 method for determining them is to calculate the two solutions of the first three 
 equations of (37) and retain only the one which satisfies the fourth relation. 
 
 Thus the invariants of the linear algebra (14) which we have found are suffi- 
 cient to isolate and characterize the generic case. 
 
 The University of Chicago, 
 Chicago, III. 
 
VITA 
 
 Cyrus Colton MacDuffee was born June 29, 1895, at Oneida, New York. 
 He attended the public grammar school and high school in that city. In 1917 
 he was graduated from Colgate University with the degree of B.S. The two 
 following years he spent as instructor in mathematics at Colgate. He attended 
 the summer school of Cornell University during the summer of 1917. He en- 
 tered the University of Chicago in June, 1919, and was in residence for nine 
 successive quarters. He attended courses under Professors E. H. Moore, 
 L. E. Dickson, G. A. Bliss, E. J. Wilczynski, H. E. Slaught, J. W. A. Young 
 and K. Laves of the University of Chicago faculty, and during the summer quar- 
 ters he attended courses under Professor M. W. Haskell of the University of 
 California, Professor S. Lefschetz of the University of Kansas, Professor T. 
 Hildebrandt of the University of Michigan, and Professor H. Blumberg of the 
 University of Illinois. In June 1920 he received the S.M. degree, and in Sep- 
 tember 1921 the degree of Ph.D. Since then he has been instructor at Prince- 
 ton University. 
 
 The writer wishes to express his gratitude to all the members of the mathe- 
 matics department, and especially to Professor L. E. Dickson under whose 
 direction this thesis was written. 
 
__J^- the lase date stamped beW. 
 
 JUN 2 1< 
 
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