Mr Bromwich, Theorems on Matrices and Bilinear Forms. 77 



invariant-factors are all linear, Darboux's rule may be at once 

 reduced to the following : — 



Let (X — c) be an a-times repeated factor of | AA' — XE | and 



write 



A — T\t /-(a-m) ^'S' /^ ( 



'r, s= 1, 2, ..., w 



this representing a determinant of (n + m) rows and columns 

 (according to a notation suggested by Frobenius 1 and Nanson 2 ) 

 where Xc rs x r y s = A A' -(c -\-t) E and the p's are arbitraries. Also 

 write £ m for the determinant obtained from A m by replacing one 

 line p x (m) , ...,.p n m with x ly ...,x n ; rj m is found similarly with 



3/i> •••> y«- 



Finally put X m = Zm/(Am-i&m) h , 



J- m == Vm/\^m—i^m) s ) 



and then we shall have 



AA' -XE=2(c-\)(X 1 Y 1 + ...+ X a Y a ), 



where the summation extends to all roots of | AA' — \E \ = 0, 

 and the products on the right are ordinary, not symbolical. 



We treat (A' A — XE) in the same way ; we should note that 

 the invariant-factors of | A' A — \E \ and of | AA' - \E \ are the 

 same. For we have 



A {A' A- XE) = (AA' -XE)A, 



and thus the invariant-factors must be the same by a known 

 theorem. Thus we shall find 



A' A - \E= 2 (c- \) (BiH, + ... + 2 a H a ), 



where c, a are the same as in the last expression. 

 Hence we have two substitutions R, S such that 



B (A A' - XE) R' = Zc r x r yr - ~xE> 



S' (A' A -XE)S = %c r x r y r - XE, 



or RR' = E = SS', 



and (RA) (RA)' = tc r x r y r = (AS)' (AS). 



Thus if we take the form 



B = 2c,$x r y r 



1 Berliner Sitzungsberichte, 1894, " Ueber das Tragheitsgesetz, etc." 



2 Phil. Mag., vol. 44, 5th Series (1897), p. 396. 



