The set of all $m;{ imes};n$ matrices with entries in $mathbb{Z}_+$ is denoted by $mathbb{M}{m{ imes}n}(mathbb{Z}_+)$. We say that a linear operator T on $mathbb{M}{m{ imes}n}(mathbb{Z}_+)$ is a (U, V)-operator if there exist invertible matrices $U;{in}; mathbb{M}{m{ imes}n}(mathbb{Z}_+)$ and $V;{in};mathbb{M}{m{ imes}n}(mathbb{Z}_+)$ such that either T(X) = UXV for all X in $mathbb{M}{m{ imes}n}(mathbb{Z}_+)$, or m = n and T(X) = $UX^{t}V$ for all X in $mathbb{M}{m{ imes}n}(mathbb{Z}_+)$. In this paper we show that a linear operator T preserves the rank of matrices over the nonnegative integers if and only if T is a (U, V)operator. We also obtain other characterizations of the linear operator that preserves rank of matrices over the nonnegative integers.
