Construction of exceptional copositive matrices

Abstract

An $n\times n$ symmetric matrix A is copositive if the quadratic form $x^{T}Ax$ is nonnegative on the nonnegative orthant $R_{\ge 0}^n$. 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 $B{B}^T$ for some $n\times r$ matrix B with nonnegative entries. The above inclusions are strict for $n\ge 5$. The first main result of this article is a free probability inspired construction of exceptional copositive matrices of all sizes ≥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 Klep and 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 complement this result by establishing an analogous bound when the sections of the cones are unit balls in the Frobenius inner product.