46 PROCEEDINGS OP THE AMERICAN ACADEMY 



V. 



ON A THEOKEM OF SYLVESTER'S RELATING TO 

 NON-DEGENERATE * MATRICES. 



By Henry Taber, Clark University. 



Presented March 9, 1892. 



In" his Memoir on Matrices, Cayley enunciated the theorem 

 that the most general matrix commutative with a given matrix was 

 a rational integral function of it. Of this theorem Sylvester gave 

 a proof, in the Johns Hopkins University Circulars, Vol. III. 

 pp. 34 and 57, for the case in which the latent roots of the given 

 matrix were all distinct, hut pointed out that, when the latent 

 roots of the given matrix were not all distinct, the theorem did not 

 always hold. Subsequently, in the Comptes Rendus, Vol. XCVIII. 

 p. 471, Sylvester stated that he had proved that Cayley's theorem 

 held for a matrix of the third order which was not degenerate (de- 

 ror/atoire),^ irrespective of equalities between its latent roots; and 

 he further stated that Cayley's theorem was probably true for non- 

 degenerate matrices of any order; i. e. that the most general ma- 

 trix commutative with a non-degenerate matrix of any order 

 whatever was a rational integral function of it. I shall in this 

 paper verify this conjecture of Sylvester's. The theorem follows 

 very simply from one given by the author in a note in these Pro- 

 ceedings for May 26, 1891,$ which contains the complete solution 

 of the problem to find the most general matrix commutative with 

 a given matrix. 



The theorem is then to be proved for the case in which the latent 

 roots of the given matrix are not all distinct. If the distinct la- 

 tent roots of the given matrix Q are g x , g 2 , g 3 , etc., of multiplicity 



* Sylvester employs the term matrice derogatoire (Comptes Rendus, Vol. 

 XCVIII. p. 471) to denote a matrix whose identical equation is of order lower 

 than the order of the matrix ; such a matrix I term a degenerate matrix. 



t See preceding note. 



t These Proceedings, Vol. XXVI p 64. 



