A random copositive matrix is completely positive with positive probability

• URL: http://arxiv.org/abs/2305.16224v2
• Date: Wed, 14 Feb 2024 13:15:47 GMT
• Title: A random copositive matrix is completely positive with positive probability
• Authors: Igor Klep, Tea \v{S}trekelj, Alja\v{z} Zalar
• Abstract summary: An $ntimes n$ symmetric matrix $A$ is copositive if the quadratic form $xTAx$ is nonnegative on the nonnegative orthant. The main result, proved using Blekherman's real algebraic geometry inspired techniques and tools of convex geometry, shows that as $n$ goes to infinity, the ratio of volume radii of the two cones is strictly positive.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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