Mr Bromwich , Theorems on Matrices and Bilinear Forms. 75 
Theorems on Matrices and Bilinear Forms. By T. J. I’A. 
Bromwich, M.A., St John’s College. 
[Received August 1900.] 
This paper consists of three parts ; in part 1 is a discussion of 
Sylvester’s rule for the biorthogonal reduction of a bilinear form, 
and a short account of former papers on the same subject. Part 2 
contains an account and comparison of various formulae used by 
different writers to evaluate functions of a matrix or bilinear form. 
Part 3 contains an investigation of the invariant-factors of any 
function of a matrix. 
1. Reduction of a bilinear form by biorthogonal substitutions. 
This problem has been considered by the following authors : 
Beltrami, Giornale di Matem., 1873, t. 11, p. 98. 
Jordan, Lioumlles Journal, 1874, t. 19 (2me serie), p. 35. 
Kronecker, Berliner Monatsberichte, 1874 (16 May) = Ges. 
WerJce, Bd. 1, p. 410. 
Cosserat, Annales de Toulouse , 1889, t. 3, pp. 1 — 12. 
Sylvester, Comptes Rendus, 1889, t. 108, pp. 651 — 653. 
„ Messenger of Math., vol. 19, pp. 1, 42. 
Jordan’s method depends on finding stationary values of the 
given form, when the variables are subject to the two conditions 
X# 2 = 1 = % 2 ; his process is finally a step-by-step method. 
Kronecker’s paper consists in the main of various criticisms on 
Jordan’s and a sketch of an alternative method. 
Cosserat considers some special cases of Jordan’s method, with 
particular consideration of the alternate form when the co- 
efficients of x r y s and of x s y r are equal and opposite in sign ; while 
the coefficient of x r y r is zero. 
Sylvester’s proof (given in the Messenger) depends on an 
infinite repetition of two infinitesimal orthogonal substitutions. 
By this means he proves the possibility of reducing the given 
bilinear form with two orthogonal substitutions. Sylvester also 
gives without proof a rule ( Comptes Rendus) for finding the two 
reducing substitutions of a given form. Here we give a short 
proof of this rule, using Frobenius’s 1 method of combining sym- 
bolically bilinear forms; or Cayley’s 2 for combining matrices. 
1 Crelle's Journal (1878), Bd. 84, p. 1. 
2 Phil. Trans. (1858), vol. 148, p. 17 ; Coll. Works, vol. 2, p. 475. 
