76 Mr Bromivich, Theorems on Matrices and Bilinear Forms. 
In Frobenius’s notation the symbolical product of two bilinear 
forms A, B is given by 
AB = 1 dAdB > 
oy r ox r 
(r = l, 2 
and it is to be remarked that the product has the same effect as 
making a linear substitution on the %’s in B or on the y’s in H. 
The accented letter A' denotes the conjugate form of A, obtained 
by interchanging the %’s and y’s in A. 
Suppose that we have 
B = BAS , 
where A is the given bilinear form, and R, S are orthogonal forms. 
Then, taking the conjugate forms (i.e. changing % r to y r and vice 
versa) we have 
B ' = S'A'R'. 
Hence BB' = R(AA') R\ 
B , B = S'(A'A)S i 
for we have R'R = 8 S' = 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 (A A') 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, A A' 
and E are real positive definite forms ; hence all the roots of 
| AA — \E | =0 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 etseq. ; Ges. Werke, Bd. 1, 
p. 233 et seq. 
2 l.e. supra; and Berl. Monatsber., 1868, p. 310= Ges. Werke, Bd. 2, p. 19. 
3 Liouville’8 Journal, 1874, t. 19, 2me s6rie, pp. 347 — 397. 
