68 Dr Baker, On the Invariant Factors of a Determinant. 



does not exist, there is a relation 



A.D^x^ + ... + AfD 1 -^^ + AD l ~*x = 0, 



in which A is not zero and may be taken unity ; we have thus a 

 set of solutions 



y = x + A x x w + . . . + AfX^ 



of the equations D l ~ 1 y=0. By the definition of the number g 

 there exists a relation 



B.D^-y^ + ...+ B g D l ^y { 9) + BD l ~ 2 y = 



in which B is not zero and may be taken unity. Hence the set 



z = y + B x yn -F . . . + B g y ( ® 



satisfies the equations D l ~ 2 z = 0. Proceeding thus we obtain a set 

 of solutions 



v = u + H 1 uU + ... + H q u® 



of the equations Dv = ; thence by the definition of the number r 

 there exists a relation 



K x v® + ... + K r vM + v=0 



which expresses v, and hence u, and so on, and hence finally 

 expresses «asa linear function of the f+g+...+r sets put down 

 in § 1, so that f+... + r is the number of linearly independent 

 sets of solutions of D l x = 0. 



In precisely the same way it can be shewn that every solution 

 of D l ~*y = is a linear function of the g + ... + q + r sets other 

 than x {1) , ..., x^ ] ; and so on ; and finally that every set of solutions 

 of Dv = Q is linearly expressible by the r sets occurring in the last 

 line of our scheme. Thus the rank of the matrix D is n — r, of 

 D 2 is n — (r + q), and so on ; and finally the rank of D l is 



n — (/+... +r). 



Retaining the notation by which all the sets in the first line 

 are denoted by the letter x, all in the second line by the lefcter 

 y, ..., all in the last line by the letter v we clearly have, arranging 

 the columns of the scheme as rows 



Dv® =0,DuU = v (1 >, ...,DxV =2/W \ 



if rows, I columns 



DvW = 0,Du^ =vW, ...,M» =y& j 



I g-f rows, 



"" ^ , , v _. . . . . f £ — 1 columns, 



Dv® =0,DuW =vW, ...,Dy {g) = z™ I 



