Mr Bromwich, Theorems on Matrices and Bilinear Forms. 83 
quote Frobenius ( Berliner Sitzungsberichte, 1896, p. 11), who, 
after defining f(A) in the way just described, says — “ In dieser 
Weise hat Stickelberger in seiner akademischen Antrittsschrift 
Zur Theorie der linearen Differentialgleichungen (Leipzig, 1881) 
die allgemeine Potenz definirt und. ..benutzt. Eine weniger 
genaue Definition giebt Sylvester Sur les puissances et les racines 
des substitutions lineaires ( Gomptes Readies, t. 94, 1882, p. 55).” 
It has not been possible for me to consult this work of Stickel- 
berger’s, but from Frobenius’s statement it seems clear that 
Stickelberger was the first author to publish a general definition 
of any power of a bilinear form or matrix 1 . Sylvester’s definition 
is less exact, because it does not allow for the possibility that 
\rE—A\ may have repeated factors; and the same objection 
applies to his definition of any function of a matrix 2 , which is 
f(A)=tf{a) 
(A^-bE){A -cE^A^-lE) 
(a — b) (a — c) ... (a — l) ’ 
where | rE — A | = (r — a) (r — b)(r — c) ... (r — l). It is easy to see 
that this definition is included in Frobenius’s as a special case ; 
but the latter seems easier for purposes of actual calculation, even 
when Sylvester’s can be applied. Sylvester has used his formula 
to find the square-root of a quaternion 3 . 
Buchheim and Taber were led independently to extensions of 
Sylvester’s formula to the case of repeated factors in | rE — A |. 
We shall now indicate how their extensions are related to 
Frobenius’s form already given. Buchheim’s result 4 seems to have 
been the first definition of any function of a matrix in the case 
when | rE — A | has repeated factors ; it is not very different from 
one of the definitions already obtained, though he determines the 
function which we have called yjr(r) by the fact that f (r) = 0 is 
the equation of lowest degree satisfied by r = A ; no method being 
given for finding f (r) when A is known. If now we write 
6 (r) = (r — a) a 
f( r ) 
fa (r) = 
f(r) 
{i — a) a ’ 
Buchheim’s formula is equivalent to 
f(A)=2f a (A) 
e(a)E + ff(a){A-aE) + ... 
6*- 1 (a) 
(a— 1 )! 
(A - aEy-' . 
1 Cayley (1858) had obtained the expression for any power of a 2-rowed matrix; 
but his method seems quite impracticable in general. 
2 Johns Hopkins TJniv. Circulars, 3 (1882), pp. 9 and 210. 
3 Phil. Mag., 5th Series, vol. 16, pp. 267 and 394; vol. 17, p. 392; vol. 18, 
p. 454 (1883—84). 
4 Phil. Mag., 5th Series, vol. 22 (1886), p. 173. 
