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

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 2 n nonzero entries; we establish that this conjecture is true for 5 × 5 sign patterns. We also establish analogous results for nonzero patterns.