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.
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.
Copyright Date
2007
Language
en
File Format
application/pdf
Recommended Citation
DeAlba, Luz M.; Hentzel, Irvin R.; Hogben, Leslie; McDonald, Judith; Mikkelson, Rana; Pryporova, Olga; Shader, Bryan; and Vander Meulen, Kevin N., "Spectrally arbitrary patterns: Reducibility and the 2n conjecture for n = 5" (2007). Mathematics Publications. 84.
https://lib.dr.iastate.edu/math_pubs/84
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.