| Title: | Construction of exceptional copositive matrices |
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| Authors: | ID Štrekelj, Tea (Author)
ID Zalar, Aljaž (Author) |
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| Files: |  PDF - Presentation file, download (809,21 KB)
MD5: 36175EA11953C9BA86E147A75FF45887
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0024379525003465
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | An $n \times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant ${\mathbb R}^n_{\ge 0}$. 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. |
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| Keywords: | copositive matrix, completely positive matrix, positive polynomial, sum of squares, convex cone |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.12.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 368-384 |
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| Numbering: | Vol. 727 |
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| PID: | 20.500.12556/DiRROS-23758  |
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| UDC: | 512:519.8 |
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| ISSN on article: | 0024-3795 |
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| DOI: | 10.1016/j.laa.2025.08.010 |
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| COBISS.SI-ID: | 251106307 |
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| Note: |
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| Publication date in DiRROS: | 01.10.2025 |
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| Views: | 224 |
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| Downloads: | 92 |
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