Centrosymmetric Stochastic Matrices

Authors

Lei Cao, Shandong Normal University - China; Nova Southeastern UniversityFollow
Darian McLaren, Brandon University - Canada
Sarah Plosker, Brandon University - Canada

Document Type

Article

Publication Date

10-29-2019

Publication Title

arXiv.org

Keywords

Stochastic matrix, Centrosymmetric matrix, Extreme points, Birkhoff theorem, Faces

First Page

1

Last Page

12

Abstract

We consider the convex set Γ_m,n of m×n stochastic matrices and the convex set Γ^π_m,n ⊂Γ_m,n of m×n centrosymmetric stochastic matrices (stochastic matrices that are symmetric under rotation by 180 degrees). For Γ_m,n, we demonstrate a Birkhoff theorem for its extreme points and create a basis from certain (0,1)-matrices. For Γ^π_m,n, 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 Γ_m,n and Γ^π_m,n, 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.

Additional Comments

NSERC Discovery grant #: 1174582, Canada Foundation for Innovation grant #: 35711; Canada Research Chairs Program grant #: 231250

NSUWorks Citation

Cao, Lei; McLaren, Darian; and Plosker, Sarah, "Centrosymmetric Stochastic Matrices" (2019). Mathematics Faculty Articles. 281.
https://nsuworks.nova.edu/math_facarticles/281