Construction of exceptional copositive matrices

Štrekelj, Tea (avtor), Zalar, Aljaž (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: copositive matrix, completely positive matrix, positive polynomial, sum of squares, convex cone
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FRI - Fakulteta za računalništvo in informatiko; FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2025
• Št. strani: Str. 368-384
• Številčenje: Vol. 727
• PID: 20.500.12556/RUL-182649
• UDK: 512:519.8
• ISSN pri članku: 0024-3795
• DOI: 10.1016/j.laa.2025.08.010
• COBISS.SI-ID: 251106307
• Datum objave v RUL: 20.05.2026

Gradivo je del revije

• Naslov: Linear algebra and its applications
• Skrajšan naslov: Linear algebra appl.
• Založnik: Elsevier
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: kopozitivna matrika, popolnoma pozitivna matrika, pozitivni polinom, vsote kvadratov, konveksni stožec