A generalization of a theorem of von Neumann

Abstract

In 1953 von Neumann proved that every \(n\times n\) doubly substochastic matrix A can be increased to a doubly stochastic matrix, i.e., there is an \(n\times n\) doubly stochastic matrix D for which \(A\le D.\) In this paper, we will discuss this result for a class of \(I\times I\) doubly substochastic matrices. In fact, by a constructive method, we find an equivalent condition for the existence of a doubly stochastic matrix D which satisfies \(A\le D,\) for all \(A\in {\mathcal {A}},\) where \({\mathcal { A}}\) is assumed to be a class of (finite or infinite) doubly substochastic matrices. Such a matrix D is called a cover of \(\mathcal {A}.\) The uniqueness of the cover will also be discussed. Then we obtain an application of this concept to a system of (infinite) linear equations and inequalities.