A short note on doubly substochastic analog of Birkhoff's theorem

TL;DR

This paper extends Birkhoff's theorem to doubly substochastic matrices, showing they can be expressed as convex combinations of a limited number of subpermutation matrices based on their structure.

Contribution

It introduces a novel decomposition of doubly substochastic matrices into convex combinations of subpermutation matrices, linked to their nonzero elements and indecomposable components.

Findings

Doubly substochastic matrices can be expressed as convex combinations of subpermutation matrices.

The number of matrices in the combination is bounded by the sum of nonzero elements and indecomposable components.

Provides a structural insight into the decomposition of doubly substochastic matrices.

Abstract

Let B be an n by n doubly substochastic matrix. We show that B can be written as a convex combination of no more than {\sigma}(B)+t subpermutation matrices, where {\sigma}(B) is the number of nonzero elements in B and t is the number of fully indecomposable components of B^{comp}, the minimal doubly stochastic completion of B obtained by a specific way.