78 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 



we have B — B' and BB' = B 2 = 2c r oc r y r , 



and on substitution 



(B-iRA) (B-iRA)' = E=(ASB- 1 )' (A SB- 1 ) 



or the two forms B^RA, A SB- 1 are orthogonal. Now the product 

 of two orthogonal forms is itself orthogonal, so as R, S are or- 

 thogonal we have 



P = B~ 1 (RAS), Q = (RAS)B~ 1 



as two orthogonal forms. 



Hence RA (SP' 1 ) = B = (Q^R) AS, 



and SP- 1 , Q-*R are both orthogonal, so the original problem has 

 been solved. But in one case it is found that we do not need to 

 calculate P, Q. To prove this we have that 



Q = BPB~ 1 



is an orthogonal form and thus Q = (Q ')~ 1 . 

 Or, since B' = B, we have 



BPB- 1 = (B-'P'B)- 1 = B-'PB, 



for P is orthogonal and so (P')- 1 = P. 



Hence B 2 P = PB 2 and B 2 = Xc r x r y r , 



and if all the coefficients c r are different, this equation can 

 only hold if P is of the form Hd r x r y r , where the d's may be any 

 arbitraries. Now P is orthogonal and so we must have 



d r 2 = l. (r=%2,.,.,n) 



In this case 



RAS=BP=2(±c,?)w r y r , 



and the form has been again reduced by the two orthogonal sub- 

 stitutions R, S. Obviously a sufficient condition (though not 

 necessary) is that all the roots of | AA' - \E\ =0 should be 

 different. In this case we have say 



AA' = 2c r X r T r , A' A = Sc,5,.H r , 



and then A = X ( ± c r i) Z, H, 



which is Sylvester's rule. 



But if the roots of | A A' — \E j = are not all different it may 

 be necessary to modify one or other set of variables 1 by a further 



1 That is, either the X's or the H's (it is immaterial which); the fact that the 

 two substitutions R, S do not necessarily reduce A seems not to have been noticed 

 before. 



