Linear Preservers of Perimeters of Nonnegative Real Matrices

• Journal title : Kyungpook mathematical journal
• Volume 48, Issue 3,  2008, pp.465-472
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2008.48.3.465
Title & Authors
Linear Preservers of Perimeters of Nonnegative Real Matrices
Song, Seok-Zun; Kang, Kyung-Tae;

Abstract
For a nonnegative real matrix A of rank 1, A can be factored as $\small{ab^t}$ 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 $\small{k\geq4}$. That is, a linear operator T preserves perimeters 2 and $\small{k(\geq4)}$ if and only if it has the form T(A) = UAV or T(A) = $\small{UA^tV}$ with some invertible matrices U and V.
Keywords
rank;perimeter;linear operator;(U, V)-operator;
Language
English
Cited by
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