The copositive range

Abstract

We consider symmetric copositive matrices $A\in \mathbf{M}_n\,(\mathbb{R})$, which by definition satisfy $x^TAx\geq 0$ for all nonzero $x\geq 0$. We introduce the notion the copositive range of a copositive matrix $A$,$$CR(A)=\{x^TAx \,:\, x\geq 0,\; \|x\|_2=1\},$$ and prove that $CR(A)$ is an interval contained in the numerical range of $A$. We focus on the properties and the endpoints of $CR(A)$, which are associated with the Pareto eigenvalues of $A$.