Linear preservers of copositive and completely positive matrices

TL;DR

This paper investigates the structure of invertible linear maps on symmetric matrices that preserve the cones of copositive and completely positive matrices, providing a complete description for the two-dimensional case.

Contribution

It characterizes the form of invertible linear maps preserving copositive and completely positive cones, especially fully describing the case for 2x2 matrices.

Findings

Complete description of invertible maps on $ ext{S}^2$ preserving $CP_2$.

Structural insights into linear preservers of copositive and completely positive cones.

Identification of conditions under which linear maps preserve these cones.

Abstract

The objective of this manuscript is to understand the structure of an invertible linear map on the space of real symmetric matrices $\mathcal{S}^n$ that leaves invariant the closed convex cones of copositive and completely positive matrices ($COP_n$ and $CP_n$). A description of an invertible linear map on $\mathcal{S}^2$ such that $L(CP_2) \subset CP_2$ is completely determined.