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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 (relative to $K_1$ and $K_2$) if $A(K_1) \subseteq K_2$. $A$ is said to be semipositive (relative to $K_1$ and $K_2$) if there exists a $x \in K_1$ such that $Ax \in K_2^\circ$ (the interior of $K_2$). We prove that $A$ is nonnegative if and only if $A+B$ is semipositive for every semipositive matrix $B$.

A Characterization of Nonnegativity Relative to Proper Cones