76 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 



In Frobenius's notation the symbolical product of two bilinear 

 forms A, B is given by 



. „ ^ dA dB , 



and it is to be remarked that the product has the same effect as 

 making a linear substitution on the x's in B or on the y's in A. 

 The accented letter A' denotes the conjugate form of A, obtained 

 by interchanging the xs and y's in A. 

 Suppose that we have 



B = RAS, 



where A is the given bilinear form, and R, S are orthogonal forms. 

 Then, taking the conjugate forms (i.e. changing x r to y r and vice 

 versa) we have 



B' = S'A'R'. 



Hence BE = R(AA') R', 



B'B = 8'(A'A)S, 



for we have R'R = SS' = E (by definition of orthogonal forms), 

 where E is the unit-form (Einheitsform), i.e. the identical or unit- 

 matrix in Cayley's and Sylvester's terminology. 



We now see that the problem of reducing A to a canonical 

 form by two orthogonal substitutions depends on reducing (A A') 

 by one orthogonal cogredient substitution and (A' A) by another. 

 We observe that both of the forms (AA') and (A'A) are sym- 

 metrical, consequently the problem is much simpler than the 

 corresponding one of reducing any bilinear form. 



To find R we have to consider the determinant 



\AA'-\E\; 



and we observe that if the coefficients of A be all real, AA' 

 and E are real positive definite forms ; hence all the roots of 

 | A A' — \E | = are real and positive ; and all the invariant- 

 factors are linear 1 . 



Sylvester's method of reduction is now seen to be equivalent 

 to that of Weierstrass 2 or Darboux 3 for reducing symmetrical 

 forms with linear invariant-factors. Sylvester assumes that all the 

 roots of | AA' — \E | =0 are distinct, but the necessary modi- 

 fication is not very great. In fact, if we remember that the 



1 Weierstrass, Berliner Monatsberichte, 1858, p. 207 et seq. ; Ges. Werke, Bd. 1, 

 p. 233 et seq. 



2 I.e. supra; and Berl. Monatsber., 1868, p. 310= Ges. Werke, Bd. 2, p. 19. 



3 Lionville's Journal, 1874, t. 19, 2me serie, pp. 347—397. 



