OP ARTS AND SCIENCES. 



65 



'1 — 



where the letters on the left and above denote the number of rows and 

 columns in the respective squares and rectangles below and to the 

 right. Similarly with respect to the square arrays 0^, etc., correspond- 

 ing to the other latent roots.* If, now, <^ is a matrix commutative 

 with Q, the most general expression for is ^ = w i^ cj~ ^, where 



c 



* This representation of a matrix, what I term its canonical or standard 

 form, was, I believe, first given explicitly by Weyr, Comptes Rendus, Vol. C. ; 

 it was, however, given substantially by Buchheim, earlier, in Proc. Lond. Math. 

 Soc, Vol. XVI. 



VOL. XXVI. (n. 8. XVIIl.) 6 



