Linear preservers of copositive matrices

Abstract

An n-by-n real symmetric matrix is called copositive if its quadratic form is nonnegative on nonnegative vectors. Our interest is in identifying which linear transformations on symmetric matrices preserve copositivity either in the into or onto sense. We conjecture that in the onto case, the map must be congruence by a monomial matrix (a permutation times a positive diagonal matrix). This is proven under each of some additional natural hypotheses. Also, the into preservers of standard type are characterized. A general characterization in the into case seems di¢ cult, and examples are given. One of them provides a counterexample to a conjecture about the into preservers.