A characterization of nonnegativity relative to proper cones

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.