A theorem on infinite positive matrices

Abstract

1. Let A = (aij) be an infinite matrix with positive elements aij > 0, i, j = 0, 1, *, (matrices (aij) with aij > 0 will be called in the sequel positive matrices). It was proved in [3 ], that (1) if A is a finite positive matrix, a unique doubly stochastic matrix T exists such that T=D1AD2 where D1 and D2 are diagonal matrices with all elements on the diagonal positive and are unique up to a scalar factor. The method used in [3], and introduced first in [4], is a constructive one and consists in alternate normalizing rows and columns of A and proving the convergence of this procedure. Another proof of (1) was given in [1]. This second proof uses besides Brouwer's fixed point theorem the fact, that (2) the set { x = (x0, x1, , xn); xi real numbers, E = 1 and x?>01 is homeomorphic to an n-dimensional ball. Although a purely existential one, this second proof contains a statement about the existence of directions of fixed points for some mapping defined by help of a finite matrix A. In this paper we note that statement (1) does not hold for infinite matrices and prove a theorem generalizing properly (1) to the case of infinite matrices. Essentially, both proofs in [1] and in [3] could be, with some nontrivial changes, applied to give the desired generalization. The difficulty in generalizing the proof given in [3 ] consists i.a. in the fact that for an infinite matrix Ej aij (or Es aij) is not always finite. The idea of our proof is similar to that of [1 ] except that (2) is not used and that Brouwer's theorem is replaced by the theorem of Schauder (see [2 ]). In the sequel a matrix A = (aij) with aij > O will be called a positive matrix and a diagonal matrix with positive diagonal elements will be called a positive diagonal matrix. Finally bij = 0, i 5j; 1, i =j, will denote the delta of Kronecker.