20.500.12556/RUL-182649 Construction of exceptional copositive matrices An $n \times n$ 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. copositive matrix completely positive matrix positive polynomial sum of squares convex cone kopozitivna matrika popolnoma pozitivna matrika pozitivni polinom vsote kvadratov konveksni stožec true false true Angleški jezik Slovenski jezik Članek v reviji 2026-05-20 09:19:22 2026-05-20 09:19:30 2026-05-21 04:57:12 0000-00-00 00:00:00 2025 0 0 Str. 368-384 Vol. 727 Dec. 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512:519.8 0024-3795 10.1016/j.laa.2025.08.010 251106307 1-s2.0-S0024379525003465-main.pdf 1-s2.0-S0024379525003465-main.pdf 1 EDF7D2D3F931835410085C963330B0FF 2807a0ddf8f927f03992b7f5ff7f5d2c2d95d68185219b4d82618c7e6c8fab94 fb6851e6-541c-11f1-b0ab-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=234099 https://www.sciencedirect.com/science/article/pii/S0024379525003465 1 eda5a157-541b-11f1-b0ab-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=234098 Fakulteta za računalništvo in informatiko Fakulteta za matematiko in fiziko 0 0 0