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Linear Preservers of Perimeters of Nonnegative Real Matrices
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  • 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;
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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) = UAV or T(A) = with some invertible matrices U and V.
rank;perimeter;linear operator;(U, V)-operator;
 Cited by
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. crossref(new window)

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

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

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

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.

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C. K. Li and S. J. Pierce, Linear preserver problems, Amer. Math. Monthly, 108(2001), 591-605. crossref(new window)

S. Z. Song and S. G. Hwang, Spanning column ranks and their preservers of nonnegative matrices, Linear Algebra Appl., 254(1997), 485-495. crossref(new window)