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 Differentialgleicliungen (Leipzig, 1881) 

 die allgemeine Potenz definirt und...benutzt. Eine weniger 

 genaue Definition giebt Sylvester Sur les puissances et les racines 

 des substitutions lineaires (Comjjtes Rendus, 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 



J ' J v ; (a — b)(a — c) ... (a— I) 



where | rE — A | = (r — a) (r — b) (r — c) . . . (r - I). 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 ^ (r) by the fact that yjr (r) = is 

 the equation of lowest degree satisfied by r = A ; no method being 

 given for finding \jr (r) when A is known. If now we write 



Buchheim's formula is equivalent to 



f(A) = ^ a (A) 



d{a)E + 8'(a){A-aE) + ... 



# a_1 («) / A 



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 Univ. 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. 



