A Random Copositive Matrix Is Completely Positive with Positive Probability

.An \(n× n\) symmetric matrix \(A\) is copositive if the quadratic form \(x^TAx\) is nonnegative on the nonnegative orthant \( Rⁿ≥ 0\) . The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the form \(BB^T\) for some \(n× r\) 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.Keywordscopositive conepositive polynomialsum of squaresconvex conecompletely positive matrixMSC codes13J3047L0752A4090C2290C27