Mr Bromwich, Theorems on Matrices and Bilinear Forms. 89 



will be of the form [r— f(a)] l+1 and (k — m) of the form [r— f(a)] 1 ; 

 where I is the quotient and (m— 1) the remainder when (a — 1) 

 is divided by k. 



In particular, if a = 1, or the invariant-factors of A to base 

 (r — a) are linear, we have (as remarked by Frobenius) linear 

 invariant-factors [r — /(a)] of f(A), whatever maybe the value 

 of k. Another special case is when a % k, and we have a linear 

 invariant-factors [r —f{a)]. Of this Frobenius has given an 

 example 1 ; in his results we have 



f(a) = 1, f (a) = =/" (a) = . . , =/«- (a), 



f(b) = 0=f(b)=...=f^(b), 



/(c) = 0=/'(c) = ...=/v-i(c). 



Hence we have here (a + a' + a" + . . . ) linear invariant-factors 

 (?• — 1); all the others being simply r. Thus the equation satisfied 

 by f(A) is 



[fW?-f(A) = 0, 



as proved by Frobenius. 



It will be observed that in case we have 



/' (a) = 0=/" (a) = ... =_/•*-(«), 



it will in general be impossible to find a function g, such that 

 </ (/(^.)) = A. For, by what has been proved, corresponding to 

 the single invariant-factor (r — a) a of A, we have k invariant- 

 factors of f(A); and thus at least k of g{f{A)); so, unless either 

 a=l or A; = 1, there will be more invariant-factors of g (/(A)) 

 than of A, and the equality is impossible 2 . 



In conclusion it may be well to point out that there is no 

 difficulty in verifying my results on the invariant-factors of f(A) 

 by direct calculation of the H.c.F. of successive minors. In fact 

 my results were originally worked out in this way ; but it 

 appeared very long to give a satisfactory description of the process, 

 while the method given above is comparatively short although 

 apparently a less direct way of attacking the problem. 



1 Berliner Sitzungsberichte, 1896, p. 601 ; the functions are those denned at the 

 end of part 2 above. 



2 Frobenius, Grelle, Bd. 84, p. 14 (Satz iv, v). 



VOL. XI. PT. I. 



