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, Oiomale di Matem., 1873, t. 11, p. 98. 

 Jordan, Liouvilles Journal, 1874, t. 19 (2me serie), p. 35. 

 Kronecker, Berliner Monatsberichte, 1874 (16 May) = Ges. 

 Werke, Bd. 1, p. 410. 



Cosserat, Annales de Toulouse, 1889, t. 3, pp. 1 — 12. 

 Sylvester, Gomptes 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 = %y- ; 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. 



