Symmetric arctic ranks of nonnegative matrices and their linear preservers

Abstract

A rank one matrix can be factored as for vectors and of appropriate orders. The perimeter of this rank one matrix is the number of nonzero entries in plus the number of nonzero entries in . A matrix of rank k is the sum of k rank one matrices, a rank one decomposition. The perimeter of a matrix A of rank k is the minimum over all rank one of A of the sums of perimeters of the rank one matrices. The arctic rank of a matrix is one half the perimeter. In this article, we characterize the linear operators that preserve the symmetric arctic ranks of nonnegative symmetric matrices.