On linear preservers of semipositive matrices

TL;DR

This paper characterizes linear transformations that preserve semipositive matrices related to proper cones, resolving a recent conjecture and extending the understanding of preservers for various cone structures.

Contribution

It provides a complete description of linear preservers of semipositive matrices for arbitrary proper cones, resolving a conjecture and linking to other linear preserver problems.

Findings

Resolved a recent conjecture on into linear preservers.

Determined linear preservers for semipositive matrices with arbitrary proper cones.

Connected strong linear preservers to other linear preserver problems.

Abstract

Given proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, an $m \times n$ matrix $A$ with real entries is said to be semipositive if there exists a $x \in K_1^{\circ}$ such that $Ax \in K_2^{\circ}$, where $K^{\circ}$ denotes the interior of a proper cone $K$. This set is denoted by $S(K_1,K_2)$. We resolve a recent conjecture on the structure of into linear preservers of $S(\mathbb{R}^n_+,\mathbb{R}^m_+)$. We also determine linear preservers of the set $S(K_1,K_2)$ for arbitrary proper cones $K_1$ and $K_2$. Preservers of the subclass of those elements of $S(K_1,K_2)$ with a $(K_2,K_1)$-nonnegative left inverse as well as connections between strong linear preservers of $S(K_1,K_2)$ with other linear preserver problems are considered.