Infinite products of analytic matrices

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: Let A(x) = [aij(x)] and B (x) = [6,-y (x)] (i,j -1, ••■, n), be 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 equationis 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 * Presented to the Society, December 28, 1915.t See, for example, Proceedings of the American Academy of Arts and Sciences, vol.49 (1913), pp.521-568; in particular p. 540.t For the elements of the theory of matrices assumed in the present paper see Schlesinger, Vorlesungen über linearen Differentialgleichungen, pp.18-19.* Questions of factorization for such matrices have been considered by Hensel and Landsberg, Theorie der algebraischen Funktionen einer Vvriabeln, etc., p. 163.