174 Theorem relating to Matrices. 



among their latent roots. In a paper on the theory of ma- 

 trices, published in vol. xvi. of the ' Proceedings of the London 

 Mathematical Society/ I have given a canonical form to 

 which matrices can be reduced. If we have a matrix m with 

 r distinct latent roots \ l . . . X n of multiplicities n 1 . . . n r , I 

 have shown that there are n t latent points answering to X i? 

 and that these can be arranged in groups, such that for one 

 of these groups (e l . . . e s ) we have 



_ X t e, x \j e 2 + ei... X, e s + g a _i 

 e x e 2 e s 



It can be shown without difficulty that if <f> is any function, 

 we have 



» m _ $\e x <f}\je 2 + (f>%e x ...<f>\e + <ft%g s -i + . ■ . + ft'^^-iei 

 e-i e 2 e s 



Now call s the length of the chain (ei . . . e s ), and let Si be the 

 length of the longest chain appertaining to Xj- ; then it is easy 

 to see that the extension of Prof. Sylvester's theorem amounts 

 to finding a function / such that 



and that these conditions are satisfied if we take 

 where 



Xi x=^-x 1 )\^x i y\. . (*-Vi) i| - , (*-W>* ,:| • • • (*-K) Sr 



and we therefore have 





It is obvious that this reduces to Prof. Sylvester's formula if 

 the latent roots are all unequal, since in this case s ly s 2 , &c. 

 are all equal to unity. 



The Grammar School," Manchester, 

 June 25, 1886. 



