A random copositive matrix is completely positive with positive probability
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916504777261056 |
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| author | Klep, Igor Štrekelj, Tea Zalar, Aljaž |
| author_facet | Klep, Igor Štrekelj, Tea Zalar, Aljaž |
| contents | An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the form $BB^T$ for some (possibly rectangular) matrix $B$ with nonnegative entries. The main result, proved using Blekherman's real algebraic geometry inspired techniques and tools of convex geometry, shows that asymptotically, as $n$ goes to infinity, the ratio of volume radii of the two cones is strictly positive. Consequently, the same holds true for the ratio of volume radii of any two cones sandwiched between them, e.g., the cones of positive semidefinite matrices, matrices with nonnegative entries, their intersection and their Minkowski sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_16224 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A random copositive matrix is completely positive with positive probability Klep, Igor Štrekelj, Tea Zalar, Aljaž Functional Analysis Optimization and Control Quantum Physics 13J30, 47L07, 52A40 (Primary), 90C22, 90C27 (Secondary) An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the form $BB^T$ for some (possibly rectangular) matrix $B$ with nonnegative entries. The main result, proved using Blekherman's real algebraic geometry inspired techniques and tools of convex geometry, shows that asymptotically, as $n$ goes to infinity, the ratio of volume radii of the two cones is strictly positive. Consequently, the same holds true for the ratio of volume radii of any two cones sandwiched between them, e.g., the cones of positive semidefinite matrices, matrices with nonnegative entries, their intersection and their Minkowski sum. |
| title | A random copositive matrix is completely positive with positive probability |
| topic | Functional Analysis Optimization and Control Quantum Physics 13J30, 47L07, 52A40 (Primary), 90C22, 90C27 (Secondary) |
| url | https://arxiv.org/abs/2305.16224 |