Functions of Limiting Matrices
F. B. Pidduck · Proceedings of the London Mathematical Society · 1921
1. Interesting problems arise in the theory of matrices when two or more of the roots are equal.This case has been discussed by Frobenius,* Sylvester,* Buchheim,t and Tabeiy § by different methods.Sylvester!!, gave the formula for any function of a matrix with unequal roots, ami suggested that the case of equal roots might be treated by passing to the limit.Sylvester's suggestion has to be modified before it can be put into practice, as the purely symbolic method leads to difficulties when we come to consider the non-scalar fractional powers of a scalar.v , § It appears that several lines of investigation can be coordinated by reverting to Grassmann's treatment of a matrix as an open product."The limiting process can then be carried out in full generality, and thus we have convenient explicit formulae in terms of the scalar coefficients of the degenerate matrix.If all the roots \ u A. 2 > ..., \ n of a matrix of order n are distinct there •are n distinct (generalised) axes u v u it ..., it n .For simplicity we shall denote Grassmann's external multiplication by simple juxtaposition as in the latter part of Al, avoiding the square brackets of A*2.Write \ \ ) , II* I ) it,,, 0