PROCEEDINGS 



OF THE 



Camkitrg* l^insnpjnal Snxieig* 



On the invariant factors of a determinant. By H. F. Baker, 

 D.Sc, St John's College. 



[Read 19 January 1903.] 



This paper gives a proof of the fundamental theorem relating 

 to the reduction of a matrix to its canonical form ; for it the 

 writer would claim only that it is complete in itself and 

 strictly elementary. It was written in July 1900 in connexion 

 with a reading of Schlesinger's treatise on linear differential 

 equations and the paper of Ed. Wehr on matrices in the first 

 volume of the Monatshefte fur Mathematih. The proof in §§ 7, 8 

 of the paper appears interesting notwithstanding its length ; of 

 the result there verified another proof is given by Netto, Acta 

 Math. t. xvii. 



§ 1. Let a be any square matrix of n rows and columns, and 

 6, 6' ... the different roots of the determinantal equation of the 

 ?ith order F (p) = \ a — p \ = 0. Then since the determinant of the 

 matrix D = a — vanishes, the n equations in the n variables 

 x x ...x n expressed by 



D l x = (a-d) l % = 0, 



wherein I is any positive integer, have a certain number of linearly 

 independent sets of solutions, in terms of which all other sets of 

 solutions can be linearly expressed. 



Let f be the greatest number of sets x (1) , oc i2) , ..., each of n 

 elements, which satisfy the equations 



Z% = 



VOL. XII. PT. II. 5 



