86 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 



This expression is of the same form as Taber's ; but his defini- 

 tions of the functions corresponding to A a , A b , A c , ... are much 

 longer than those given above and are superficially, at any rate, 

 very different. Taber states 1 that his functions have the same 

 characteristic properties, namely 



A a * = A a , A b * = A b> ... 



A a A b = 0, A b A e = 0, ... 



A a + A b + A c +...=E, 



and this seems to justify the assumption that the two sets of 

 functions are really the same. It should, however, be observed 

 that, in Taber's final result, the indices are not the same as our 

 a, 13, 7, ... ; but are the indices of the factors (? — a), (r — b), 

 (r — c), ... in <f) (r) = | rE — A \. These indices will be in general 

 greater than a, 0, y, ... and their use may lead to the retention of 

 various terms in the value for f(A) which actually vanish ; for, we 

 have seen that A a (A — aE) 1 = whenever l>{a — 1). This 

 modification of Taber's form arises from the fact that, although 

 <f)(A) = 0, yet in general <£ (r) = is not the equation of lowest 

 degree satisfied by r = A. 



3. The invariant-factors of any function of a bilinear form. 



Frobenius has proved (Grelle, Bd. 84, pp. 24, 25) that if 

 f (r) 4= at r = a, b, c, ..., then f(A) has invariant-factors 2 

 [r— f(a)] a , [r —f(b)] p , ... corresponding to the ones (r — a) a , 

 (r — by, ...of A. I have had, however, occasion to calculate the 

 invariant-factors for some functions which do not satisfy these 

 conditions ; and it seems as if a note on the general theory of such 

 a case might be of interest. 



Suppose then that for r = a we have 



f(a) = 0,f'(a)=0, ...,/*-* (a) = 0, 



but /*(«) + 0; 



we shall investigate the invariant-factors off(A) which correspond 

 to the single one (i — a) a of A. 



1 For the proofs of these properties, Taber refers to a paper of his, which I have 

 not been able to find. An account of Taber's other papers may be found in the 

 Clark University Decennial Volume (Worcester, Mass. 1899) where there is, however, 

 no reference to Stickelberger's or Buchheim's papers. 



2 Here by the invariant-factors of a bilinear form is meant those of the charac- 

 teristic determinant of the form; e.g. the invariant-factors of A imply those of 

 \rE-A\. 



