Construction of exceptional copositive matrices

TL;DR

This paper constructs exceptional copositive matrices of all sizes greater than or equal to five using free probability techniques and analyzes the asymptotic ratio of volume radii of sections of copositive and completely positive matrix cones.

Contribution

It introduces a free probability inspired method to construct exceptional copositive matrices for all sizes ≥ 5 and extends previous volume ratio bounds to Frobenius norm sections.

Findings

Constructed exceptional copositive matrices for all sizes ≥ 5.

Extended volume ratio bounds to Frobenius norm sections.

Provided asymptotic analysis of cone section volume radii.

Abstract

An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$. The cone of copositive matrices contains the cone of matrices which are the sum of a positive semidefinite matrix and a nonnegative one and the latter contains the cone of completely positive matrices. These are the matrices of the form $BB^T$ for some $n\times r$ matrix $B$ with nonnegative entries. The above inclusions are strict for $n\geq5.$ The first main result of this article is a free probability inspired construction of exceptional copositive matrices of all sizes $\geq 5$, i.e., copositive matrices that are not the sum of a positive semidefinite matrix and a nonnegative one. The second contribution of this paper addresses the asymptotic ratio of the volume radii of compact sections of the cones of copositive and completely positive matrices. In a previous work by the authors, it was shown that, by identifying symmetric matrices naturally with quartic even forms, and equipping them with the $L^2$ inner product and the Lebesgue measure, the ratio of the volume radii of sections with a suitably chosen hyperplane is bounded below by a constant independent of $n$ as $n$ tends to infinity. In this paper, we extend this result by establishing an analogous bound when the sections of the cones are unit balls in the Frobenius inner product.