# Algebraic and Geometric Properties of $\mathcal{L}^n_+$-Semipositive Matrices and $\mathcal{L}^n_+$-Semipositive Cones

## TL;DR

This paper explores algebraic and geometric properties of matrices that are semipositive with respect to the Lorentz cone, providing new characterizations and conditions, especially for diagonal and orthogonal matrices, and analyzing related cones' structures.

## Contribution

It introduces new algebraic and geometric characterizations of $\, ext{L}^n_+$-semipositive matrices, including conditions for diagonal and orthogonal cases, and studies the structure of associated cones.

## Key findings

- Necessary and sufficient algebraic conditions for $\, ext{L}^n_+$-semipositive matrices.
- Characterizations of diagonal and orthogonal $\, ext{L}^n_+$-semipositive matrices.
- Conditions under which the cones $\, ext{K}_{A, ext{L}^n_+}$ and $\, ext{S}_{A, ext{L}^n_+}$ are ellipsoidal.

## Abstract

Given a proper cone $K$ in the Euclidean space $\mathbb{R}^n$, a square matrix $A$ is said to be $K$-semipositive if there exists an $x\in K$ such that $Ax\in \text{int}(K)$, the topological interior of $K$. The paper aims to study algebraic and geometrical properties of $K$-semipositive matrices with special emphasis on the self-dual proper Lorentz cone $\mathcal{L}^n_+=\{x\in \mathbb{R}^n:x_n\geq 0,\sum\limits_{i=1}^{n-1}x_{i}^2\leq x_n^2\}$. More specifically, we discuss a few necessary and other sufficient algebraic conditions for $\mathcal{L}^n_+$-semipositive matrices. Also, we provide algebraic characterizations for diagonal and orthogonal $\mathcal{L}^n_+$-semipositive matrices. Furthermore, given a square matrix $A$ and a proper cone $K$, geometric properties of the semipositive cone $\mathcal{K}_{A,K}=\{x\in K:~Ax\in K\}$ and the cone of $\mathcal{S}_{A,K}=\{x:Ax\in K\}$ are discussed in terms of their extremals. As $\mathcal{L}^n_+$ is an ellipsoidal cone, at last we find results for the cones $\mathcal{K}_{A,\mathcal{L}^n_+}$ and $\mathcal{S}_{A,\mathcal{L}^n_+}$ to be ellipsoidal.