Abstract
Given a proper cone K in the Euclidean space Rn, a square matrix A is said to be K-semipositive if there exists an x∈K such that Ax∈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 L+n={x∈Rn:xn≥0,∑i=1n−1xi2≤xn2}. More specifically, we discuss a few necessary and other sufficient algebraic conditions for L+n-semipositive matrices. Also, we provide algebraic characterizations for diagonal and orthogonal L+n-semipositive matrices. Furthermore, given a square matrix A and a proper cone K, geometric properties of the semipositive cone KA,K={x∈K:Ax∈K} and the cone of SA,K={x:Ax∈K} are discussed in terms of their extremals. As L+n is an ellipsoidal cone, at last we find results for the cones KA,L+n and SA,L+n to be ellipsoidal.
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