54 PROCEEDINGS OF THE AMERICAN ACADEMY 



if the canonical forms of O and Cl 1 are respectively w 6 w -1 and 

 g/0w' -1 , and provided \p is the most general matrix commutative 

 with 6. 



This solution of the equation Cl <j> = <£ Cl', as also the ahove 

 mentioned solution of the equation O </> = c/> O, depends upon the 

 reduction of a matrix to its canonical form. In the assumed 

 canonical form of Cl, viz. co 6 to -1 , the matrix 6 is dependent solely 

 upon the characteristics of the latent roots of Cl. Having deter- 

 mined the latent roots of Cl and their characteristics, if one value 

 of co is known, the solution of the preceding equation Cl t/> = <£ 12' 

 gives in terms of w every solution of the prohlem to reduce a given 

 matrix Cl to its canonical form. For if Cl' = 0, the most general 

 solution of the equation 



reduces to 



(f) = Oil}/, 



where ij/ is the most general matrix commutative with 6, and w is 

 some particular value of </> satisfying the equation Cl = <£ 6 </> -1 . 

 But since the solutions of this equation are all included in those of 

 the equation Cl <f> = <f> 6, if the constants in if/ are so taken that its 

 determinant (and consequently the determinant of <£) is not null, 

 (j> = w ij/ is the most general solution of the problem to reduce a 

 matrix to its canonical form. 



If O and CI' are matrices of the same kind,* and such that for 

 any latent root g the increments of nullity of successive powers 

 of CI — g (or CI' — g) are severally equal to the nullity of O — g 

 (or of CI' — g), until a power is reached whose nullity is equal to 

 its vacuity, then the most general solution of the equation 

 CI <f> = <£ Ci' can he expressed in terms of certain arbitrary matrices 

 and of products of powers of CI and CI' less their latent roots. Thus, 



let the latent roots of O (and of CI') be g t , g 2 , g t , respectively 



of multiplicity m x , m 2 , .... m ( ', and suppose that the nullities of 

 ^ — ffi> ^ — <7 2 > etc. are m\, m' 2 , etc. From the supposed equality 

 in the increments of nullity of CI — g lf etc., it follows that 



m i in 2 Mi 



are all integers. Let, now, 



* That is, fl and n' are such that it is possible to find a matrix x t0 satisfy 

 the equation CI' = x G X" '• 



