Linear Preservers of Perimeters of Nonnegative Real Matrices
Seok-Zun Song, Kyung-Tae Kang · Kyungpook mathematical journal · 2008
For a nonnegative real matrix A of rank 1, A can be factored as ab t for some vectors a and b.The perimeter of A is the number of nonzero entries in both a and b.If B is a matrix of rank k, then B is the sum of k matrices of rank 1.The perimeter of B is the minimum of the sums of perimeters of k matrices of rank 1, where the minimum is taken over all possible rank-1 decompositions of B. In this paper, we obtain characterizations of the linear operators which preserve perimeters 2 and k for some k ≥ 4. That is, a linear operator T preserves perimeters 2 and k(≥ 4) if and only if it has the form T (A) = U AV , or T (A) = U A t V with some invertible matrices U and V .