A characterization of nonnegativity relative to proper cones

TL;DR

This paper characterizes nonnegativity of matrices relative to proper cones in terms of semipositivity, providing a new equivalence and exploring its applications in cone theory.

Contribution

It establishes that a matrix is nonnegative relative to cones if and only if adding any semipositive matrix yields a semipositive matrix, offering a novel characterization.

Findings

A matrix is nonnegative iff its sum with any semipositive matrix is semipositive.

The paper provides applications of this characterization in cone theory.

New insights into the structure of nonnegative matrices relative to proper cones.

Abstract

Let $A$ be an $m \times n$ matrix with real entries. Given two proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, we say that $A$ is nonnegative if $A(K_1) \subseteq K_2$. $A$ is said to be semipositive if there exists a $x \in K_1^\circ$ such that $Ax \in K_2^\circ$. We prove that $A$ is nonnegative if and only if $A+B$ is semipositive for every semipositive matrix $B$. Applications of the above result are also brought out.