The Positive Semidefinite Rank of Matrices and Polytopes

Abstract

The positive semidefinite (psd) rank of a nonnegative <italic>p</italic> × <italic>q</italic> matrix <italic>M</italic> is defined to be the smallest integer <italic>k</italic> such that there exist <italic>k</italic> × <italic>k</italic> psd matrices <italic>A</italic>₁,…, <italic>A</italic>_p and <italic>B</italic>₁,…, <italic>B</italic>_q with <italic>M</italic>_ij = < <italic>A</italic>_i , <italic>B</italic>_j >. The psd rank of a polytope is defined to be the psd rank of its slack matrix, a special matrix encoding the vertex-facet structure of the polytope, which is intimately related to representations of the polytope as a projection of a slice of a closed convex cone. We apply tools from algebraic geometry, optimization, linear algebra, and other fields to gain a more complete understanding of psd rank. We present lower and upper bounds on the psd rank of polytopes and investigate the class of polytopes achieving the lower bound. We classify the set of all slack matrices and show computational complexity results relating to psd rank. We explore the space of possible psd factorizations as a topological space and exhibit a special case where it is connected.