Campus Units

Mathematics

Document Type

Article

Publication Version

Accepted Manuscript

Publication Date

6-2007

Journal or Book Title

Linear Algebra and its Applications

Volume

423

DOI

10.1016/j.laa.2006.12.018

Abstract

A sign pattern Z (a matrix whose entries are elements of {+, −, 0}) is spectrally arbitrary if for any self-conjugate spectrum there is a real matrix with sign pattern Z having the given spectrum. Spectrally arbitrary sign patterns were introduced in [J.H. Drew, C.R. Johnson, D.D. Olesky, P. van den Driessche, Spectrally arbitrary patterns, Linear Algebra Appl. 308 (2000) 121–137], where it was (incorrectly) stated that if a sign pattern Z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern, then Z is a spectrally arbitrary sign pattern, and it was conjectured that the converse is true as well; we present counterexamples to both of these statements. In [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary patterns, SIAM J. Matrix Anal. Appl. 26 (2004) 257–271] it was conjectured that any n ×n spectrally arbitrary sign pattern must have at least 2n nonzero entries; we establish that this conjecture is true for 5 × 5 sign patterns. We also establish analogous results for nonzero patterns.

Comments

This is a manuscript of an article from Linear Algebra and its Applications 423 (2007): 262, doi:10.1016/j.laa.2006.12.018. Posted with permission.

Rights

This manuscript version is made available under the CCBY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Copyright Owner

Elsevier Inc.

Language

en

File Format

application/pdf

Published Version

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