LINEAR OPERATORS THAT PRESERVE PERIMETERS OF MATRICES OVER SEMIRINGS

Abstract

A rank one matrix can be factored as <TEX>$\mathbf{u}^t\mathbf{v}$</TEX> for vectors <TEX>$\mathbf{u}$</TEX> and <TEX>$\mathbf{v}$</TEX> of appropriate orders. The perimeter of this rank one matrix is the number of nonzero entries in <TEX>$\mathbf{u}$</TEX> plus the number of nonzero entries in <TEX>$\mathbf{v}$</TEX>. 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.