Mr Bromwich , Theorems on Matrices and Bilinear Forms. 89 
will be of the form [r and (k — m) of the form [r — f(a)] 1 ; 
where l is the quotient and (m— 1) the remainder when (a — 1) 
is divided by k. 
In particular, if a = 1, or the in variant- factors of A to base 
(r — a) are linear , we have (as remarked by Frobenius) linear 
invariant-factors \r — /(a)] of f(A), whatever may be 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, /' (a) = 0 =/"(«)=... =/«-' (a), 
f(b) = 0 -/' (b) = ...=p-Hh), 
/(c) = 0=/'(c) = ...=/r-(c). 
Hence we have here (a + a 7 -fa" + ...) linear invariant-factors 
(r — 1); all the others being simply r. Thus the equation satisfied 
by/(^) is 
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 
g (f(A)) = 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 k = 1, there will be more invariant-factors of g(f(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 defined at the 
! end of part 2 above. 
2 Frobenius, Crelle, Bd. 84, p. 14 (Satz iv, v). 
VOL. XI. PT. I. 
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