Considering copositivity locally

We say that a symmetric matrix $A$ is copositive if $\mathbf{v}^T A\mathbf{v}\geq0$ for all nonnegative vectors $\mathbf{v}$. 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 $\delta > 0$ satisfying $A + \delta B$ 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 $C$.