Linear mappings preserving the copositive cone

Abstract

Let ${\mathcal{S}}_n$ be the set of all $n$-by-$n$ symmetric real matrices, and let ${\mathcal{C}}_n$ be the copositive cone, that is, the set of all matrices $a\in {\mathcal{S}}_n$ that fulfill the condition $u^{\top }au⩾0$ for all $n$-vectors $u$ with nonnegative entries. We prove that a linear mapping $\phi :{\mathcal{S}}_n\to {\mathcal{S}}_n$ satisfies $\phi ({\mathcal{C}}_n)={\mathcal{C}}_n$ if and only if $$\phi (x)=m^{\top }xm$$ for a fixed monomial matrix $m$ with nonnegative entries.

5 pages