Linear mappings preserving the copositive cone

TL;DR

This paper characterizes all linear transformations on symmetric matrices that preserve the copositive cone, showing they are precisely those conjugations by monomial matrices with nonnegative entries.

Contribution

It provides a complete characterization of linear maps preserving the copositive cone, identifying them as conjugations by specific monomial matrices.

Findings

Linear maps preserving the copositive cone are characterized.

Such maps are exactly conjugations by monomial matrices with nonnegative entries.

The result offers a structural understanding of copositive cone automorphisms.

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 $u^\top a u\geqslant0$ for all $n$-vectors $u$ with nonnegative entries. We prove that a linear mapping $\varphi:\mathcal{S}_n\to \mathcal{S}_n$ satisfies $\varphi(\mathcal{C}_n)=\mathcal{C}_n$ if and only if $$\varphi(x)=m^\top xm$$ for a fixed monomial matrix $m$ with nonnegative entries.