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Mathematical Problems in Engineering
Volume 2009, Article ID 301582, 25 pages
http://dx.doi.org/10.1155/2009/301582
Research Article

Nonnegativity Preservation under Singular Values Perturbation

1Departamento de Matemรกticas, Universidad Catรณlica del Norte, Casilla 1280. Antofagasta, Chile
2Departamento de Matemรกtica y Fรญsica, Universidad de Magallanes, Casilla 113-D, Punta Arenas, Chile

Received 30 December 2008; Accepted 18 May 2009

Academic Editor: David Chelidze

Copyright © 2009 Emedin Montaño et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study how singular values and singular vectors of a matrix ๐ด change, under matrix perturbations of the form ๐ด+๐›ผ๐ฎ๐‘–๐ฏโˆ—๐‘– and ๐ด+๐›ผ๐ฎ๐‘๐ฏโˆ—๐‘ž, ๐‘โ‰ ๐‘ž, where ๐›ผโˆˆโ„, ๐ด is an ๐‘šร—๐‘› positive matrix with singular values ๐œŽ1โ‰ฅ๐œŽ2โ‰ฅโ‹ฏโ‰ฅ๐œŽ๐‘Ÿ>0,๐‘Ÿ=min{๐‘š,๐‘›}, and ๐ฎ๐‘—,๐ฏ๐‘˜,๐‘—=1,โ€ฆ,๐‘š;๐‘˜=1,โ€ฆ,๐‘›, are the left and right singular vectors, respectively. In particular we give conditions under which this kind of perturbations preserve nonnegativity and certain matrix structures.