Centrosymmetric Stochastic Matrices

TL;DR

This paper characterizes the extreme points and bases of convex sets of stochastic matrices, including centrosymmetric ones, using graph-theoretic methods and basis constructions, extending classical results like Birkhoff's theorem.

Contribution

It extends Birkhoff's theorem to centrosymmetric stochastic matrices, providing explicit basis constructions and characterizations of extreme points based on matrix symmetry and graph properties.

Findings

Characterization of extreme points of stochastic matrices.

Basis construction for centrosymmetric stochastic matrices.

Analysis of bipartite graph properties related to these matrices.

Abstract

We consider the convex set $\Gamma_{m,n}$ of $m\times n$ stochastic matrices and the convex set $\Gamma_{m,n}^\pi\subset \Gamma_{m,n}$ of $m\times n$ centrosymmetric stochastic matrices (stochastic matrices that are symmetric under rotation by 180 degrees). For $\Gamma_{m,n}$, we demonstrate a Birkhoff theorem for its extreme points and create a basis from certain $(0,1)$-matrices. For $\Gamma_{m,n}^\pi$, we characterize its extreme points and create bases, whose construction depends on the parity of $m$, using our basis construction for stochastic matrices. For each of $\Gamma_{m,n}$ and $\Gamma_{m,n}^\pi$, we further characterize their extreme points in terms of their associated bipartite graphs, we discuss a graph parameter called the fill and compute it for the various basis elements, and we examine the number of vertices of the faces of these sets. We provide examples illustrating the results throughout.