Linear mappings preserving the copositive cone

Shitov, Yaroslav

doi:10.1090/proc/15432

Linear mappings preserving the copositive cone
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by Yaroslav Shitov PDF

Proc. Amer. Math. Soc. 149 (2021), 3173-3176 Request permission

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 $u^\top a u\geqslant 0$ 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 \begin{equation*} \varphi (x)=m^\top xm \end{equation*} for a fixed monomial matrix $m$ with nonnegative entries.

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