A new certificate for copositivity

In this article, we introduce a new method of certifying any copositive matrix to be copositive. This is done through the use of a theorem by Hadeler and the Farkas Lemma. For a given copositive matrix this certificate is constructed by solving finitely many linear systems, and can be subsequently checked by checking finitely many linear inequalities. In some cases, this certificate can be relatively small, even when the matrix generates an extreme ray of the copositive cone which is not positive semidefinite plus nonnegative. This certificate can also be used to generate the set of minimal zeros of a copositive matrix. In the final section of this paper we introduce a set of newly discovered extremal copositive matrices.