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;

Song, Seok-Zun; Kang, Kyung-Tae;

Abstract

For a nonnegative real matrix A of rank 1, A can be factored as 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 . That is, a linear operator T preserves perimeters 2 and if and only if it has the form T(A)

Keywords

rank;perimeter;linear operator;(U, V)-operator;

Language

English

References

1.

L. B. Beasley, G. S. Cheon, Y. B. Jun and S. Z. Song, Rank and perimeter preservers of Boolean rank-1 matrices, J. Korean Math. Soc., 41(2004), 397-406.

2.

L. B. Beasley, D. A. Gregory and N. J. Pullman, Nonnegative rank-preserving operators, Linear Algebra Appl., 65(1985), 207-223.

3.

L. B. Beasley and N. J. Pullman, Boolean rank-preserving operators and Boolean rank-1 spaces, Linear Algebra Appl., 59(1984), 55-77.

4.

L. B. Beasley, S. Z. Song and S. G. Lee, Zero term rank preservers, Linear and Multilinear Algebra, 48(2001), 313-318.

5.

L. B. Beasley, S. Z. Song, K. T. Kang and B. K. Sarma, Column ranks and their preservers over nonnegative real matrices, Linear Algebra Appl., to appear.

6.

A. Berman and R. J. Plemmons, Nonegative matrices in the mathematical sciences, Academic, New York (1976).