Mathematics Faculty Articles

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

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

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

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