Infinite products of analytic matrices

Abstract

In a large part of the theory of functions'of a single complex variable the matrix of analytic functions rather than the single analytic function must be taken as the fundamental element.This is certainly the case for the functions defined by linear difference and differential equations.The goal of the present paper is to show that the classical results of Weierstrass and Mittag-Leffler, treating of the formation of infinite products of functions with assigned singularities, admits of a natural extension to infinite products of matrices.The matrices considered will be square matrices of m2 elements and of determinant not identically zero.The concept of equivalence, which I have developed elsewhere, t lies at the basis of this extension: Letbe two matrices of analytic functions, each analytic in the neighborhood of a point x = Xo but not necessarily analytic at the point x0.If the matrix M ( x ) defined by the matrix equation A(x) = M(x)B(x) is composed of elements m»y ( x ) each analytic at x = xo and if the determinant | M ( x ) | of the matrix M ( x ) is not zero at x0, then A ( x ) is equivalent to B (x) at a: = x0 (from the left)4From this definition it follows immediately that if A ( x ) is equivalent to B ( x ) at x = xo, then B ( x ) is also equivalent to A ( x ) ; if, further, B ( x ) is equivalent to 0 ( x ), then A ( x ) is equivalent to 0 ( x ).Two matrices equivalent to each other at a point have essentially the same type of singularity at the point.It is this fact which gives importance to the notion.Evidently any two matrices of functions analytic at x0 are equivalent if the determinants of the two matrices do not vanish at the point.For this reason a point for which the elements of a matrix are analytic, and the determinant

Locations

• Transactions of the American Mathematical Society - View - PDF