Doubly stochastic arrays with small support

TL;DR

This paper investigates the structure of doubly stochastic matrices with minimal support, providing solutions for all dimensions and characterizing extremal matrices, especially when the dimensions are coprime.

Contribution

It determines the smallest support doubly stochastic matrices for all dimensions and characterizes extremal matrices, highlighting the case when dimensions are coprime.

Findings

Minimum support matrices are extremal in convexity.

Examples of extremal matrices not of minimum support are provided.

When dimensions are coprime, extremal matrices are exactly those with minimum support.

Abstract

An $n \times m$ non-negative matrix with row sum $m$ and column sum $n$ is called doubly stochastic. We answer the problem of finding doubly stochastic matrices of smallest posible support for every $1 <n \leq m$. Any matrix of minimum support is extremal in the sence of convexity, while examples of extremal matrices that are not of minimum support are given. But when $n,m$ are coprime integers extremal matrices are precisely those of minimum support.