A Random Copositive Matrix Is Completely Positive with Positive Probability
Igor Klep, Tea Štrekelj, Aljaž Zalar · SIAM Journal on Applied Algebra and Geometry · 2024
Abstract. An [Formula: see text] symmetric matrix [Formula: see text] is copositive if the quadratic form [Formula: see text] is nonnegative on the nonnegative orthant [Formula: see text]. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the form [Formula: see text] for some [Formula: see text] matrix [Formula: see text] with nonnegative entries. The main result, proved using Blekherman’s real algebraic geometry inspired techniques and tools of convex geometry, shows that asymptotically, as [Formula: see text] 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.