Spectrally arbitrary patterns: Reducibility and the 2n conjecture for n = 5

Date
2007-06-01
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Hentzel, Irvin
DeAlba, Luz
Hentzel, Irvin
Hogben, Leslie
Hogben, Leslie
McDonald, Judith
Mikkelson, Rana
Pryporova, Olga
Shader, Bryan
Vander Meulen, Kevin
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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.

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<p>This is a manuscript of an article from <em>Linear Algebra and its Applications </em>423 (2007): 262, doi:<a href="http://dx.doi.org/10.1016/j.laa.2006.12.018" target="_blank">10.1016/j.laa.2006.12.018</a>. Posted with permission.</p>
Keywords
Sign pattern, Nonzero pattern, Spectrally arbitrary sign pattern, Reducible sign pattern, Irreducible sign pattern, Potentially nilpotent
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