Construction of exceptional copositive matricesŠtrekelj, Tea (Avtor) Zalar, Aljaž (Avtor) copositive matrixcompletely positive matrixpositive polynomialsum of squaresconvex coneAn $n \times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant ${\mathbb R}^n_{\ge 0}$. 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 $n\ge 5$. The first main result of this article is a free probability inspired construction of exceptional copositive matrices of all sizes $\ge 5$ 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.20252025-10-01 09:26:32Neznano23758UDK: 512:519.8ISSN pri članku: 0024-3795DOI: 10.1016/j.laa.2025.08.010COBISS_ID: 251106307sl