OP ARTS AND SCIENCES. 53 



POSTSCRIPT. 



Worcester, Mass., March 26, 1892. 



I find that the solution of the matrical equation fi<£ = <£fi given 

 in a note in these Proceedings, May 26, 1891, and upon which 

 the proof of the theorem given above is based, gives the complete 

 solution of the more general matrical equation O c/> = (f> O' for that 

 case in which it may be assumed that the determinant of <f> is not 

 null. For this case the latent roots of ft' are identical with those 

 of ft, and have, respectively, the same characteristics, i. e. the nulli- 

 ties of successive powers of ft' less any latent root g are respectively 

 equal to the nullities of successive powers of ft — g. Therefore, 

 if the canonical form of ft is to uT 1 , the canonical form of ft' is 

 w a/ -1 .* Matrices thus related are termed by Weyr matrices of 

 the same kind. 



If, then, in the equation 



ft <f> = cf) ft , 

 ft and ft' are replaced by their equivalents, it becomes 



CO 6 (o —1 <f> = <}> to' 6 co' — 1 , 



that is, 



6 . w>~ ! c/> <o' = co — * cf> to' . 0. 



If, now, we put 



the equation becomes 



\tf = CO (fi CO , 



6 if, = if/ 6. 



But the most general solution of this equation is given in the note 

 above referred to; the most general matrix commutative with 6 is 

 there denoted by -q. Therefore, the most general solution of the 

 matrical equation 



<£ ft = ft' <£ 

 is 



c£ = co if/ to' - , 



* For explanation of the term canonical form of a matrix, see paper referred 

 to, these Proceedings, Vol. XXVI p. 64. 



