A Characterization of Nonnegativity Relative to Proper Cones

Abstract

Let A be an m × n matrix with real entries. Given two proper cones K1 and K2 in ℝn and ℝm, respectively, we say that A is nonnegative if A(K1) ⊆ K2. 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.