A rank one matrix can be factored as $mathbf{u}^tmathbf{v}$ for vectors $mathbf{u}$ and $mathbf{v}$ of appropriate orders. The perimeter of this rank one matrix is the number of nonzero entries in $mathbf{u}$ plus the number of nonzero entries in $mathbf{v}$. A matrix of rank k is the sum of k rank one matrices. The perimeter of a matrix of rank k is the minimum of the sums of perimeters of the rank one matrices. In this article we characterize the linear operators that preserve perimeters of matrices over semirings.