Preprint : Considering Copositivity Locally ∗

We say that a symmetric matrix A is copositive if vTAv ≥ 0 for all nonnegative vectors v. The main result of this paper is a characterization of the cone of feasible directions at a copositive matrix A, i.e., the convex cone of symmetric matrices B such that there exists δ > 0 satisfying A + δB being copositive. This cone is described by a set of linear inequalities on the elements of B constructed from the so called set of (minimal) zeros of A. This characterization is used to furnish descriptions of the minimal (exposed) face of the copositive cone containing A in a similar manner. In particular, we can check whether A lies on an extreme ray of the copositive cone by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition for the irreducibility of A with respect to a copositive matrix C.