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

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 ${\mathcal{L}}_{+}^n=\{x\in {\mathbb{R}}^n:x_n\ge 0,{\sum }_{i=1}^{n-1}{x}_i^2\le x_n^2\}$. 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.