Symmetric arctic ranks of nonnegative matrices and their linear preservers

A rank one matrix can be factored as uv(t) for vectors u and v of appropriate orders. The perimeter of this rank one matrix is the number of nonzero entries in u plus the number of nonzero entries in v. 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.