Almost Bijective Parametrization of $3 \times 3$ Copositive Matrices

Abstract: In this work we take a deep dive into the cone of copositive $3 \times 3$ matrices. In doing so we visualize the cone, make geometric observations about it, and prove them. We then use these observations to parametrize the set. In the process we run into issues with surjectivity, and overcome them by resolving singularities and slightly shifting our original approach. We do all of this to ultimately arrive at a novel parametrization of the $3 \times 3$ copositive matrix cone which is surjective and almost injective, which we call almost bijective.