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^be •mntvereiti? ot Cbita^o 
 
 QUIVALENCE AND REDUCTION OF PAIRS 
 OF HERMITIAN FORMS 
 
 A DISSERTATION 
 
 BMITTBD TO THK FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN 
 CANDIDACY FOR THE DEGREE OP DOCTOR OF PHILOSOPHY 
 DEPARTMENT OF MATHEMATICS 
 
 HV 
 
 MAYME IRWIN LOGSDON 
 
 n^ 
 
 Private Edition, Distributed By 
 
 The University of Chicago Libraries 
 
 Chicago, Illinois 
 
 Repritited from , 
 
 Ambrican Journal of Mathematic^s 
 Vol. XLIV, No. 4, October, 1<)22 
 
Ztbe 1Ilntver0tt^ ot Cbi'cadii :.. ii* '0 1^ '{.: i v. 
 
 EQUIVALENCE AND REDUCTION OF PAIRS 
 OF HERMITIAN FORMS 
 
 A DISSERTATION ' 
 
 SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN 
 
 CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 
 
 DEPARTMENT OF MATHEMATICS 
 
 BY 
 
 MAYME IRWIN LOGSDON 
 
 M 
 
 Private Edition, Distributed By 
 
 The University of Chicago Libraries 
 
 Chicago, Illinois 
 
 Reprinted from 
 
 American Journal of Mathematics 
 
 Vol. XLIV, No. 4. October, 1922 
 
U 
 
 eXCHAlvuv. 
 
Reprinted from The American Journal op Mathematics, Vol. XlAV, Wo. 4,'6ct6b'erl 'i9^ 
 
 EQUIVALENCE AND REDUCTION OF PAIRS OF HERMITIAN 
 
 FORMS.* 
 
 By Mayme Irwin Logsdon. 
 
 Introduction and Summary. — Of fundamental importance in the theory 
 of matrices and forms has been the use by Weierstrass, Kronecker, and 
 Frobenius of the theory of elementary divisors in the study of equivalence 
 and reduction of pairs of bilinear and pairs of quadratic forms. 
 
 In this paper a generalization is made in that the basal theorems of the 
 theory are extended to any hermitian X-matrix, i.e., a matrix whose elements 
 are polynomials of degree w in X with coefficients in the field of complex 
 numbers and are such that the conjugate of the element in the ith row and 
 jth column is equal to the element in the jth row and ^th column for 
 i,j= 1, 2, "■,n. 
 
 Inasmuch as a linear substitution with matrix P on the variables of an 
 hermitian form with matrix a gives a form with matrix b = P'aP where P 
 means the matrix formed from P by taking the conjugate imaginary of 
 each element and P' means the transposed matrix P, the extension of the 
 general theory to the hermitian X-matrix is justified by the proof in Part I of 
 
 Theorem II . If h = p'aq where p and q are non-singular and inde- 
 pendent of X, and where a and b are hermitian \-matrices, then there exists a 
 matrix P such that b = P'aP. 
 
 The special application is made to hermitian X-matrices whose elements 
 are linear in X. Such will be the matrix of the pencil of forms 
 
 n n 
 
 \A — B = ^Y^ Q<aij — bij)xiXj. 
 1=1 y=i 
 
 The coincidence of the elementary divisors is found to be a necessary 
 and sufficient condition for the equivalence of two pairs of hermitian 
 matrices free of X and for the equivalence of two pairs of hermitian forms. 
 
 In Part II, the Weierstrass reduction is shown to hold in case one of 
 the forms is definite, a condition which insures reality of all the elementary 
 divisors; in fact the Weierstrass method can be used for finding the con- 
 tribution to the canonical form of any real elementary divisor. In the 
 case however of conjugate complex elementary divisors, (X — d)' and 
 (X — a)*, it was found necessary and possible to regularize the X-matrix 
 with respect to the two conjugate imaginary linear factors simultaneously 
 
 * Presented to the Society at Chicago. March 25, 1921. 
 
 247 
 
 52nr)i):i 
 
248i i ** •** . *.**.• • Lqqsd^n: Pairs of Hermitian Forms. 
 
 and also to expand the terms representing the adjoint form with respect to 
 these two factors simultaneously. 
 
 In the actual work of reduction use was made of an algebraic simplifica- 
 tion suggested and described by Dickson* in " Pairs of Bilinear or Quadratic 
 Forms." The importance of this simplification is that in the case of bilinear 
 forms the reduction may be accomplished rationally while in the case of 
 quadratic or hermitian forms the computations are simplified if the con- 
 stants c^ which appear are held until the last step of the reduction before 
 being absorbed in the variables. 
 
 The canonical form obtained f is given in Part I, Theorem V. 
 
 Part I — Equivalence. 
 We seek the conditions for equivalence of two pairs of hermitian forms : 
 
 n n n n 
 
 A = / , / J dijXiXj, A = / J / J OiiiXiXj, 
 
 i=i }=i t=i y=i 
 
 n n n n 
 
 5 = S £ bijXiXj, 5* = X) Z) ^IjXiXj, 
 1=1 y=i 1=1 i=i 
 
 with the respective matrices a, a*, b, b* of which b and b* are non-singular, 
 and where a,y = aji, etc., for i,j= 1, • • •, n. 
 
 If a linear transformation with non-singular matrix, c, whose elements 
 are independent of X, 
 
 n 
 
 Xi = J^Cijyj {i= 1, •■•,n), 
 i=i 
 
 with the induced transformation on the conjugate variables, 
 
 n 
 
 Xi = S CijVj (i = 1, ■••,n), 
 i=i 
 
 be applied to A and B, the transformed hermitian forms will have the 
 respective matrices c'ac and c'bc. If, now, these are to be the forms A* 
 and B*, we must have 
 (1) a* = c'ac, b* = c'bc. 
 
 We shall show that a necessary and sufficient condition that equations (1) 
 be satisfied is that the two hermitian X-matrices, m = a — \b and m* 
 = a* — Xb*; have the same elementary divisors. We state the problem 
 thus: 
 
 . * Transactions A. M. Society, v. 10, 1909, p. 350. 
 
 t In the Proceedings of the London Mathematical Society, v. 32, 1900, pp. 321 , 
 
 Bromwich obtains such a reduction by a special device, stating that "apparently this 
 method (the Frobenius-Weierstrass method) cannot be extended so as to cover the analogous 
 theory for conjugate imaginary substitutions, which would be applied to a pair of Hermite's 
 forms." 
 
LoGSDON : Pairs of Hermitian-'Fhrj^^ ; . . . . ••.... 24$ 
 
 Given any two hermitian \-matrices, m and m*, toith elements polynomials 
 in \, to find necessary and sufficient conditions for the existence of a non- 
 singular matrix c untk elements independent of X, such that m* = c'mc. 
 
 In proof we must show that if two hermitian X-matrices are equivalent, 
 the corresponding hermitian forms may be obtained, the one from the 
 other, by a non-singular transformation on the variables. The first step 
 of the proof will consist in establishing 
 
 Theorem I. If two hermitian \-matrices, m=a—\b and m*=a*— X&*, 
 are equivalent,* there exist two non-singular matrices, t and q, whose elements 
 are independent of X, such that 
 
 (2) m* = tmq. 
 
 Proof: By the equivalence of m and m* there exist non-singular X-matrices 
 (q and Qo with determinants free of X, such that 
 
 (3) m* = tomqo. 
 
 Now divide ^o by m* and (qo)~^ by m f in such a way as to get matrices 
 ti, t, Si, s which satisfy the relations 
 
 (4) to = m*ti + t, (qo)-^ = Sim + s, 
 
 t and s being matrices whose elements do not involve X. From (3) we get 
 torn = m*q^^. Substituting here from (4) we have " 
 
 (5) m*{ti — Si)m = m*s — tm. 
 
 Now the right member of (5) is a X-matrix of at most the first degree, while 
 for ti — Si y>^ the left member would be of at least the second degree. 
 Hence ti = Si and 
 
 (6) m*s = tm. 
 
 Whence if we knew that s (and likewise t from (6)) were non-singular, 
 we could write 
 
 (7) m* = tms~^ 
 
 and the theorem would be proved. 
 
 We proceed to show that s is non-singular. Substitute in the identity 
 
 I = QoQ6~^ for qo~^ from (42) and get 
 
 (8) / = qoSiW. H- qos. 
 Now divide qo by m* in such a way as to get 
 
 (9) qo = qm* + q, 
 
 * Two X-matrices, m and m*, are called equivalent (,B6cher, Introduction to Higher 
 Algebra, p. 274) if there exist X-matrices to and ^o each having as determinant a number not 
 zero independent of X such that m* — Umqo. 
 
 t B6cher, p. 278. 
 
250 ;.'.•' .••; : L(JP^&n: Pairs of Hermitian Forms. 
 
 where g is a matrix with elements free of X. Substituting this value in (8) 
 we have 
 
 / = q^Sim + qim*s + qs 
 
 which, by use of (6), may be written 
 
 (10) I — qs= (qoSi + qit)m. 
 
 Since the left member does not contain X we must have qoSi + qit identically 
 zero, and therefore 
 
 (11) / = qs. 
 
 s is then non-singular, and we may write (6) in the form * 
 
 m* = tmq. 
 
 Since p = t' is evidently a X-matrix whose determinant equals the con- 
 jugate of the determinant t, we may express the definition of equivalence 
 in the following form which is more convenient for hermitian forms: 
 
 Two hermitian X-matrices, m and m*, are equivalent if there exist non- 
 singular matrices p and q with determinants free of X, such that m* = p'mq. 
 
 The second step in solution of the original problem makes possible the 
 extension of the theory of equivalence to the corresponding forms. It 
 consists in proving 
 
 Theorem II. If b = p'aq, where p and q are non-singular and inde- 
 pendent of X, and where a and b are hermitian \-matrices, then there exists a 
 matrix P such that 
 
 b = P'aP 
 
 and such that P depends not on a or b but solely on p and q. 
 We have by hypothesis 
 
 (1) b = p'aq, 
 whence, since a and b are hermitian, 
 
 b = q'ap. 
 Equating these two values of b we get 
 
 (2) q'ap = p'aq, 
 
 from which 
 
 (3) {q'r'p'a = apq-\ 
 
 If now we set U = {q')~^p', then U' will be pq~^ and (3) becomes 
 
 (4) Ua = aU. 
 
 * This theorem holds if m and m* have elements of degree p in X. 
 
LoGSDON : Pairs of Hermitian'Forjiis^ 
 
 .251 
 
 t/2a = aU'''; 
 
 From this we get at once 
 
 (5) 
 
 and, in general, 
 
 (6) 
 
 From a = a and (4), (5), (6) by using any set of arithmetical multipliers 
 we get 
 
 (7) x{U)a = ax(u'), 
 
 where x(^ is any polynomial in U. Thus 
 
 (70 a = lx(U)T'axm. 
 
 Now we choose the polynomial x(^ = ^ so that V^ = U and so that V is 
 non-singular.* We have then from (7') 
 
 (8) 
 
 Substituting (8) in (1) we get 
 
 (9) 
 
 a = y-^aV. 
 
 b = p'V-^aV'q. 
 
 Now set P = V'q, then P' = q'V and h = P'aP, as desired. For, from 
 the definitions of U and V we have 
 
 U=V''= {q')-9 
 
 from which we easily obtain q'V = p' V~'^. 
 
 These theorems permit the use of the general theory of X-matrices. 
 We state the theorems and definitions which are needed in the sequel: 
 
 I. If a and h are equivalent hermitian X-matrices of rank r, and if 
 Z),(X) is the greatest common divisor of the i-rowed determinants {i ^ r) of 
 a, then it is also the greatest common divisor of the i-rowed determinants 
 of b. 
 
 II. Every hermitian X-matrix of order n and rank r can be reduced by 
 elementary transformations f to the normal form 
 
 Ei(X) 
 E2{\) 
 
 
 
 
 ^r(X) 
 
 
 
 where the coefficient of the highest power of X in each of the polynomials 
 EiQC) is unity, and E,(X) is a factor of Ei+i(K) for i = 1, 2, • • •, r — 1. 
 III. The greatest common divisor of the i-rowed determinants of an 
 
 * F is in fact a polynomial in U of degree less than n. See B6cher, I.e., p. 299. 
 t Elementary transformations as defined in B6cher, I.e., p. 262. 
 
253..-..' .";.« L(>pSDON: Pairs of Hermitian Forms. 
 
 hermitian X -matrix of rank r, when i < r, is 
 
 Di(K) = EME2(K) • • • Ei(K), 
 
 where the E's are the polynomials of the last theorem. 
 
 IV. A necessary and sufficient condition that two hermitian X-matrices 
 of order n be equivalent is that they have the same rank r, and that for 
 every value of / from 1 to r inclusive, the i-rowed determinants of one 
 matrix have the same greatest common divisor as the i-rowed determinants 
 of the other. 
 
 From the definition of the D's in III, we see that 
 
 (i= 1,2, ...,r), (Z)o(X))= 1). 
 
 Hence since the D's with the rank form a complete system of invariants, 
 since the D's completely determine the E's as w^ell as the elementary 
 divisors, and since conversely the D's are completely determined by the E's 
 or by the elementary divisors, we may state thus the 
 
 Fundamental Theorem: A necessary and sufficient condition that two 
 hermitian \-matrices be equivalent is that they have the same rank and that the 
 elementary divisors of one be identical respectively with the elementary divisors 
 of the other. 
 
 Definition: Two pairs of hermitian matrices a, b and a*, b* with elements 
 free of X will be called equivalent if there exist two non-singular matrices 
 p and q, also with elements not involving X, such that 
 
 (1) a* = Jag, b* = p'bq. 
 
 From this definition. Theorem I and the fundamental theorem we have 
 
 Theorem III. 7/ a, b and a*, b* are two pairs of hermitian matrices 
 
 independent of X, and if b and b* are non-singular, a necessary and sufficient 
 
 condition that these two pairs of matrices be equivalent is that the two \-matrices 
 
 m = a — \b, m* = a* — Xb*, 
 
 have the same elementary divisors. 
 
 Referring now to Theorem II, equivalence conditions for two pairs of 
 matrices may be stated as follows : 
 
 Theorem IV. If a, b, a*, b* are hermitian matrices independent of X, 
 and if b and b* are non-singular, a necessary and sufficient condition that a 
 non-singular matrix P exist such that 
 
 a* = P'aP, b* = P'bP, 
 
 is that the matrices a — 'Kb and a* — X6* have the same elementary divisors. 
 
Logsdon: Pairs of Hermitfan-'Fotvii.:' *..: ••.*. •*.• •/.\253 
 
 If in particular b* = b = I, where / is the unit matrix, we have 
 
 /= PP 
 
 which defines an orthogonal hermitian matrix. 
 
 Corollary: If the characteristic matrices of a and a* have the same 
 elementary divisors there will exist an orthogonal matrix P such that 
 
 a* = P'aP or a* = P-^aP, 
 
 i.e., a* is the transform of a by the orthogonal matrix P, 
 
 We have thus obtained the desired conditions for equivalence of two 
 
 pairs of hermitian forms as stated in the first paragraph of Part I. 
 
 If the matrix of the form B is the unit matrix, referring to the last 
 
 corollary we see that a transformation on the variables with orthogonal 
 
 matrix P will transform the form B into B* also with unit matrix. The 
 
 X-matrices 
 
 a - \I, a* - X/ 
 
 are now the characteristic matrices of the forms A and A*, and as before 
 a necessary and sufficient condition for the equivalence of the forms under 
 orthogonal transformation is the coincidence of the elementary divisors of 
 the characteristic matrices of the forms. 
 
 If 5 is a non-singular definite form, a preliminary transformation will 
 transform it to the sum of hermitian squares, X!i^»^t with unit matrix, and 
 since the roots of the determinant | a — X6 1 =0 are now the roots of the 
 characteristic equation of a which are known to be always real,t we have 
 the 
 
 Corollary: The elementary divisors of the pencil of hermitian forms 
 A — \B, where B is non-singular definite, are all real and of the first degree. | 
 
 As in the quadratic case we have a further 
 
 Corollary:* If A and B are hermitian forms and B is non-singular, a 
 necessary and sufficient condition that it be possible to reduce A and B 
 simultaneously by a non-singular transformation to forms into which only 
 square terms (hermitian squares) enter is that all the elementary divisors 
 of the pair of forms be of the first degree. 
 
 Finally, if both matrices are singular but at least one matrix of the 
 pencil Xifl + ^2b is non-singular, we may proceed as in the quadratic case § 
 and obtain the desired canonical form. 
 
 The canonical form. 
 
 * See B6cher, I.e., p. 30.5, for the corresponding theorem for quadratic forms. 
 t G. Kowalewski-Einfuhrung in die Determinanten Theoric, p. 130. 
 X The proof in B6cher, I.e., p. 170, for quadratic forms is applicable here. 
 § Muth, I.e., p. 87. 
 
254;/. : '.•'. ;.••{..* ••traQSTBONy Pairs of Hermitian Forms. 
 
 Theorem V. If Ci, Cg, •••, c/ are any real constants including zero, 
 equal or unequal, if ag, an, • • • , ttr are any complex numbers, equal or unequal, 
 and if e\, e^, • • • , -er are positive integers such that ^1 + ^2+ • • • + ^/ + 2^^ 
 + • • • + 2er = n, there exist pairs of hermitian forms in n variables, the 
 first form being non-singular, which have the elementary divisors 
 
 (X - ci)«s (X - C2y\ '■-, (X - c/)% (X - a,)^ (x - a,y% • • •, 
 
 (X - drY^ (X - arY-. 
 In proof after setting ^1 + ^2+ • • • + e/ = e we exhibit the forms 
 
 ei ei + ej 
 
 ■4 = ^ XiXei—i+1 -\- 2^ XiX2ei+e2—i+l + ' ' ' 
 1=1 i=ei + l 
 
 e «+2e^ 
 
 "h 2-^ XiX2e-i+l + 2^ XjX23+2eg-j+l 
 i=e—ej+l ^-e+1 
 
 n 
 
 + • • • + Z^ XjX2n—2s^-j+l- 
 
 ei ei+e2 
 
 B = zL ClXiXei-i+l + 22 (^2^i^2ei+e^i+l + ' * ' 
 
 i=l i=ei+l 
 
 e ei— 1 ei+€2— 1 
 
 "1" Z^ CfXiXze—i+l + Z-J -^i^ei-i + 2_/ -^i-^aei+ej-i 4" * " ' 
 e+e,, e+2eg 
 
 ~^ Z^ f^gX jX^e+^e^—j+l + 2-j ^g-^j-^2e+2e^—j+l + " * * 
 J=e+l J=e+eg+l 
 
 n—e^ n 
 
 + 2^ arXjX2n-2e^-j+l ~\~ ^ dfX jX2n—2e—j+\ 
 j=n-2e^+\ i=»-e^+l 
 e+2€g-l »— 1 
 
 + ZL XjX2e+2e^-j + * * * + ZL ^j^2n-2e^-]- 
 
 J=e+1 j=n-2er+l 
 
 Part II — Reduction. 
 
 In the attempt to apply the naethods of Weierstrass to reducing a pair 
 of hermitian forms: 
 
 n n n n 
 
 -4 = X) Zl ttijXiXj and ^ = Z) S bijXiXj, 
 i=i j=i 1=1 ^=1 
 
 where da = aji, ba — bju \aij\ 9^ 0, no trouble arises in finding the con- 
 tribution to the canonical form due to any real linear factor of the X-matrix, 
 Xa — h, though proof is needed that certain theorems are actually extensible 
 to this type of matrix. In dealing with complex linear factors however an 
 essential modification is required. We shall first indicate the main steps 
 in the process of reduction, then study in detail the separate cases where 
 difference of treatment is required. Wherever the Weierstrass treatment 
 as given in Muth's "Theorie und Anwendung der Elementartheiler " is valid 
 without separation of the two cases, the details of the work will be omitted 
 
Logsdon: Pairs of HermiUmcr/jrJiiii' * .: ••.; •*.• * .*,* 255 
 
 and reference to Muth given. The following notations and definitions 
 will be used : 
 
 S = determinant of the form C = \A — B. 
 
 l^ = exponent of the linear factor (X — r) in D^(K), i.e., in the greatest 
 common divisor of all the K-rowed minor determinants of S. There will 
 be at least one K-rowed minor determinant of S which contains (X — r) 
 exactly l^ times and is then defined to be regular with respect to this factor. 
 We have 
 
 and also 
 
 where (X — r)'* is an elementary divisor. 
 
 Sij = cof actor of the element Xajy — bij in S. 
 
 S(«) = principal minor determinant with n — k rows obtained from S 
 by deleting the first k rows and the first k columns. We note S^*^ = S^J-''~^\ 
 For K = 0, we define S(°> = S, and for k = n, S^"> = 1. 
 
 Sp = the principal p-rowed minor determinant in the upper left-hand 
 corner of the matrix. 
 
 Sp; ik = (p + l)-rowed minor determinant obtained by bordering S^ by 
 the ith. row and the Arth column of the original matrix. 
 
 The main steps in the reduction are: 
 
 (1) Any hermitian X-matrix may by elementary transformations * be 
 regularized with respect to 
 
 (1) any real linear factor, (X — c). 
 
 (2) any pair of conjugate imaginary linear factors, 
 
 (X — a), (X — a). 
 * Elementary transformations of an hermitian X-matrix are defined as follows: 
 
 1st. Interchanging two columns and the same two rows. 
 
 2d. Multiplying the elements of a column by a number m independent of X and then 
 multiplying the elements of the corresponding row by the conjugate of m. 
 
 3d. Adding to the elements of the Ath column the products of the corresponding 
 elements of the jth column (J 9-^ k) hy a polynomial, ^(X), then adding to 
 the elements of the A;th row the products of Jiie corresponding elements of the 
 jth row each multiplied by the conjugate, f (X), of the polynomial ^(X). 
 
 The elementary transformations of the variables of the corresponding hermitian form which 
 effect on the matrix of the form the above defined transformations are: 
 
 .^ 
 
 1st. Xi = Pi (i = 1, • • •, n; i j-^j, i 7^ k) 
 
 Xi = Vk 
 
 Xk = Vi. 
 2d. Xi = yi (i = 1, •'•,n; i 9^ j) 
 
 Xi = myi. 
 3d. Xi = yi (i = 1, • • •, n; i 9^ j) 
 
 X, = Vi + H^)yk. 
 
256 'V' J ';/] '/\ '.,', '''S^OGsi^'r' Pairs of Hermitian Forms. 
 
 Definition: A matrix S is regular with respect to a real linear factor or 
 with respect to a pair of conjugate imaginary linear factors if each of the 
 principal minor determinants obtained from S by deleting the first k rows 
 and columns, {k = 1, 2, • • •, n — 1), is so. 
 
 (2) The adjoint form may be written as a sum of terms in which every 
 factor in a denominator is regular with respect to a given linear factor 
 (or pair of factors) ; viz., 
 
 (1) 2-. g ^J^i gg, i- g,g. + • • • -h ^(„-i)^(„) • 
 
 (3) 1st. Each term on the right of (1) may be expanded with respect 
 to a real linear factor, (X — c), the total coefficients of 1/(X — c)^, 1/(X — c) 
 secured; finally, the total coefficients of 1/X^, 1/X secured. This will give 
 the contribution of this particular linear factor, X — c, to the canonical form. 
 
 2d. Each term on the right of (1) may be expanded with respect 
 to X — a, X — a simultaneously and the total coefficients of 1/X^, 1/X 
 secured. 
 
 (4) The adjoint form may be expanded by determinantal methods in 
 descending powers of X and the coefficients of 1/X^, 1/X obtained. These 
 prove to be B and A respectively. 
 
 (5) A comparison of the results of (3) and (4) with Theorem V of Part I 
 gives the desired expression for A and B. 
 
 Proofs: 
 
 (7i). Any hermitian X-matrix, S, may by elementary transformations 
 be regularized with respect to a real linear factor, X — c. 
 
 In proof we must show 
 
 (a) Every regular p-rowed minor determinant (p > 1) contains at least 
 one regular (p — 1) -rowed minor determinant as first minor. 
 
 (6) Every regular (p — l)-rowed minor determinant (p > 2) is con- 
 tained in at least one regular p-rowed minor determinant as first minor. 
 
 (c) If a (p — l)-rowed principal minor, Sp_i, is regular, but no p-rowed 
 principal minor, *Sp_i; <, u containing it is regular, there is an elementary 
 transformation of the variables which without disturbing the regularity of 
 any Sa: (^ = 1, • • • , p — 1) will so transform the matrix S that there will 
 be a regular p-rowed principal minor containing Sp..i as a first minor. 
 
 For proof of (a) and (6) see Muth, I.e., pp. 7-11. 
 
 Proof of (c): The existence of a regular p-rowed minor, iSp_i; j^ ,,, con- 
 taining Sp-i, is guaranteed by (6) above. Now apply to the variables of 
 the form a preliminary transformation which will interchange the pth and 
 jth rows and columns and the (p + l)st and kih rows and columns. We 
 now have Sp_i. p_p+i for our regular p-rowed minor. Now apply the trans- 
 
Logsdon: Pairs of Hermitinjt Ponnif' *•.• l'\ :'•: :/•'•, 257 
 
 formation 
 
 Tm: Xi = yi {i = 1, • • •, n; i =}= p + 1), 
 Xp+i = 2/p+i - TTiy^. 
 
 The effect on the matrix is to subtract the products by m of the elements 
 of the (p + l)st column from the elements of the pth column and then the 
 products by m of the elements of the (p + l)st row from the elements of 
 the pth row. Obviously Sk is not changed {k = 1, •••, p— 1), but the 
 principal minor, call it S'p, of order p and containing Sp_i, is now regular. 
 For we have 
 
 Now Sp and Sp_i;p4.i, p+i each by hypothesis contains X — c more than /p 
 times; iSp_i;p, p+i contains X — c exactly Zp times; Sp_i.p+i^p is the con- 
 jugate of Sp_i.pp+i and therefore contains the real linear factor X — c 
 exactly l^ times. It remains then to show that the sum R = wSp_i.p p+i 
 + wSp_i.p p+i does not have a higher power of X — c as factor than each 
 part, for every m. Divide /? by (X - c)'". i? = (X - c)^''[mf(X) + 7n/(X)], 
 and, since /(c) is not zero, if we set m = l/fic) we have mf{c) + mf(c) not 
 zero. Thus S^ contains X — c exactly Ip times and is regular, as stated. 
 
 (72) Any hermitian X-matrix, S, may by elementary transformations 
 be regularized with respect to an imaginary linear factor. When this is 
 done the matrix will be also regular with respect to the conjugate imaginary 
 linear factor. 
 
 As before, (a) and (6) may be assumed for the factor X — a from the 
 proof for bilinear forms. To prove (c) we apply the same transformation, 
 Tm, and get as before the right member of equation (3). Remembering 
 that the first and last terms are principal minors and hence have real 
 coefficients, and that neither is regular with respect to (X — a) (X — a), we 
 may factor the right member of (3) thus: 
 
 [(X - a)(X - a)]'''[(X - a)(X - a)J/i(X) - m(X - a^MX) 
 
 - m (X - a)f2(X) + mml(\ - a)(X - a)JUOC), 
 
 where r > 0, * > 0, p ^ 0, /2(a) 9^ 0. 
 
 If p 5^ 0, the theorem is proved since X — a is a factor of all the terms 
 in the brackets but one and consequently is not a factor of the sum. Thus 
 S'p is regular, as stated. If p = 0, we must show that mJ<i,(}C) + riiJ^QC) is 
 not divisible by X — a where /2(a) 5^ and /2(a) 9^ 0, i.e., we must show 
 that m can be so chosen that mf^ia) + fnf(a) 9^ 0. This is the same 
 condition reached before and is satisfied by m = l/fiia). Hence we may 
 use the Jacobi transformation of the adjoint form, viz.,* 
 
 ,0, lASij. X'X'r'X". . J<»>z(«) 
 
 i,j 
 
 S ' • SS' S'S" 5(n-i)5(») 
 
 * Muth, I.e., pp. 70-72. 
 
258 ,'•','?•'''''♦•'' iJacSDoN f ' Pairs of Hermitian Forms . 
 
 with a determinant regular with respect to any Hnear factor, and where 
 
 X' = SnUi + S21U2 + 8ziUz + • • • + 8nlUn 
 X" = S22U2 + S32W3 + • • • + Sn2Un 
 
 (4) r" = s'z'^m + • • • + s'n'^un 
 
 (3) To decompose the general term on the right of (3) with respect to 
 the real linear factor, X — c, we may write 
 
 where C^ is the coefficient of the highest power of X in S'^''~^^S^''\t It is 
 evident that q^, a polynomial in X with real coefficients, will not contain 
 X — c as a factor since S^*~^^ and S^"^ are regular, and we may then expand 
 X'^'^^lq and X'^'^^Jq in power series in X — c. Also, since in the definition of 
 ^w in (4) i\^Q coefficient S^,""*^ of u^ is regular with respect to X — c, Z^''^ and 
 consequently Z^*^ will contain X — c exactly l^ times. Thus 
 
 — = (X - c)'f Xa + (X - c)X^2 + (X - c)2Z.3 +•••], 
 
 ^ = (X - c)'lZa + (X - c)Z.2 + (X - cYX^z +.••]• 
 Hence 
 
 ^('<)X(«) 1 1 — ■ — 
 
 gc^D^w = C, (x-c)^'- ^"^^'"^ (X- c)Z,2H ][Za + (X - c)Z,2 + • • -1 
 
 where the Z^,^ are linearly independent polynomials in c, u^, • • • , Un and 
 the coefficients of the two forms A and B and are homogeneous in the w's. 
 Thus dropping the subscript, k, we may write the right member 
 
 ^(X^T^t^^i + Z.X\ - c) + Z3(X - c)2 + . . .] 
 
 ~ ^ C(X"^^2 -^ (7(x _ c) "^ " '• 
 
 Now, defining i'^, = Ze = Ll=iZ,Z,_,+i and G, = Z^-i = i:\=i-^XiX,-i 
 {G^ = for e^ = 1) we have 
 
 a(X- c) C,(X-c)2 c,Lx X' 
 
 
 * For convenience in notation h shall henceforth represent the exponent of the factor 
 X — c In the greatest common divisor of the (n — K)-rowed minor determinants of S. 
 We have then the inequalities, Zo ^ ^1 = ?2 = • • • = h,, with U-i — /« = e*. 
 
 t See Dickson, I.e. 
 
Logsdon: Pairs of Hermitiah'I^Ofm^X *•• '*•* :*•: .•/•*2o9 
 
 There will be a similar expression obtained from each of the terms on the 
 right of (3). The total contribution due to the real factor X — c to the 
 canonical form is then obtained by taking the sum 
 
 
 where / is the number of distinct real linear factors. The numbers 1/C, 
 may be now absorbed in the variables by x' = xj-^C^ since the original 
 forms and the transformations used have allowed irrationalities as well as 
 imaginaries. 
 
 If now the above process of regularization and expansion be repeated 
 for the remaining real linear factors, the total contribution of these factors 
 is obtained. 
 
 To decompose with respect to (X — a)(X — a) the general term of (3), 
 we need to get the partial fractions of this term which have linear or 
 quadratic denominators, viz., (X — a), (X — a), (X — of, (X — o)^ 
 (X — a)(X — a), since these and only these terms in the decomposition will 
 contribute to the coefficient of 1/X and 1/X^. We set 
 
 S(^i)5W ciQ, - a) (X - a)]'«-i+'«g2 > 
 
 where q^ is a polynomial in X containing neither X — a nor X — a as a factor 
 and with unity for the coefficient of the highest power of X. Now expand 
 X'^'^lq in powers of X — a and Z^^V? in powers of X — a. Thus 
 
 — = [(X - «)(X - a)]^l^a + (X - a)Z«2 + (X - afX^z + • • • J 
 
 — = [(X - a)(X - a)]^f Za + (X - a)Z.2 + (X - ayX^z +•••]. 
 
 where the Z^^ are polynomials in a, a, the coefficients of A and B, and 
 homogeneous in w^, • • • , w„, while the Z,^ are the corresponding conjugate 
 functions of w», ••-,«„. We have then 
 
 W=^^^ = C:(X-ar(X-arL^'" + ^^ " ^^^'^ + ^ " ^^'^'"^ 
 
 + • • -I^.i + (X - a)X^, + (X - a)2Z,3 +•••]. 
 
 Omitting all k subscripts, we may write the right member 
 
 1 r Xi 
 
 cL(X-a 
 
 Z2 . . Z 
 
 + •••+, '-z-\- Xe+i + Z^2(X - a) + 
 
 Y (X - d)*^! X - d 
 
260/.'.: ;'; /*I'.* 'i&OGsi'o^t* Pairs of Hermitian Forms. 
 
 + Z2e(X - a)-i + 
 
 ;[(-x^+-+.-^+^-+ 
 
 + X2.(X- a)^' + 
 
 Xe 
 
 y ' ■ X- o 
 
 1 rXiX2e(X - a)-i , XiZ2e-i(X - tt)^' . X^X^e-iiX " a)"-^ 
 
 — n\o-l 
 
 c\_ _ (K-ay "^ (X -ay "^ ix - a) 
 
 j^ A 2X26-2 (X — a)*~^j^ , XeXe+i , XeXe 
 
 ■I K ^T^Zl h • • ■ -T r ~ -T 
 
 (X-d)"-^ _ X-a (X-a)(X-o) 
 
 , Xe+lZe , , X2eZi(X — O)^^ . ~| 
 
 -r ;: i- • • • + 
 
 X-a (X-a)' J 
 
 ~ PX L-^l-^2e + ^'2X26-1 -|- • • • -j- XeXe+1 + ' • ' + XzeXiJ^ 
 
 + ii[a(ZiX2e + • • • + XeXe+l) + a(Ze+lZ« + • • • + X2eXt) 
 ~r -3riA2e_l + X2X2e— 2 + ' " * H" X2e_iXiJ. 
 
 Thus we have found the part of the coefficient of 1/X and 1/X^ due to 
 one term in the right member of (3) and contributed by the pair of linear 
 factors (X — a)(X — a) where the e in the last expression is the e^ which 
 belongs to the elementary divisors (X — a)% (X — a)*". The total coeffi- 
 cient of 1/X is seen to be (after the constant C^ is absorbed in the variables) 
 
 2ei 
 
 22 XjX2e^-j+l 
 J = l 
 
 and the total coefficient of 1/X^ is 
 
 «t — 2e^ 
 
 2_^ aXjX2e^-j+l + 2_/ ^XjX2e^-j+L- 
 J=l i=ek+l 
 
 For other pairs of complex elementary divisors we proceed as here, 
 then adding the coefficients of 1/X obtained from all the linear factors, real 
 and imaginary, and adding the coefficients of 1/X^ obtained from all the 
 Hnear factors, real and imaginary, we compare with the coefficients of these 
 same powers of X obtained by expanding the adjoint form 
 
 ^■■^ Kj -ij Uj (4/% 
 
 by determinanta methods, * thus obtaining the desired canonical forms 
 
 given in Part I, Theorem V. 
 
 The University of Chicago, 
 May, 1921. 
 
 * Muth, I.e., p. 81. 2 g ^x'^x^^'"- 
 
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