CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 233 



To take an example, let r = 2 and a t = 3, a 2 = 3, a 3 2. Then 

 S 1A = <? 4 ^= <? 7 ^= (4- 2, 3, 5, 6, 8); d 2u = <? 5w = (u = 3, 6, 8) 

 *AA> - ^-i ^'-i (^, ^ f - 2, 3, 5, 6, 8). 

 Setting i]o u = y u , we find that H has the following form 1 ): 



87 



Its determinant is readily seen to equal 



77 



In the general case, H is seen to take the form: 



r 



01 



f 



02 



f 



03 







l 21 











d" 



If e^ = a 2 , d'= d fli+11 and 



and 



. If %> 2 , we have 



! fll+ i = 2 fll+1 = - 



i _ 2 fll+1 = 0. Finally, if a l <a i9 we have 



1) ^17, ^47 1 ^83 1 ^86 are zero i bein ^ 6( l ual to *" ^58 1 ^2 1 ^75 respectively. 



