A random copositive matrix is completely positive with positive probability
| Autor: | Klep, Igor, Štrekelj, Tea, Zalar, Aljaž |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 27 pages. A major rewrite with the addition of applications and detailed discussions on size comparison of cones. The part with the construction of examples will appear in a separate paper |
| Databáze: | arXiv |
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