Counterexamples on spectra of sign patterns

TL;DR

This paper introduces a new technique to analyze the spectral properties of sign patterns, disproving a long-standing conjecture by providing counterexamples where superpatterns lose spectral arbitrariness.

Contribution

The authors develop a novel method surpassing the Nilpotent Jacobian approach, enabling the proof of spectral arbitrariness in previously unresolved cases and solving open problems.

Findings

Disproved a conjecture from 2000 about superpatterns and spectral arbitrariness.

Provided examples of sign patterns where superpatterns are not spectrally arbitrary.

Introduced a new technique for analyzing spectral properties of sign patterns.

Abstract

An $n\times n$ sign pattern $S$, which is a matrix with entries $0,+,-$, is called spectrally arbitrary if any monic real polynomial of degree $n$ can be realized as a characteristic polynomial of a matrix obtained by replacing the non-zero elements of $S$ by numbers of the corresponding signs. A sign pattern $S$ is said to be a superpattern of those matrices that can be obtained from $S$ by replacing some of the non-zero entries by zeros. We develop a new technique that allows us to prove spectral arbitrariness of sign patterns for which the previously known "Nilpotent Jacobian" method does not work. Our approach leads us to solutions of numerous open problems known in the literature. In particular, we provide an example of a sign pattern $S$ and its superpattern $S'$ such that $S$ is spectrally arbitrary but $S'$ is not, disproving a conjecture proposed in 2000 by Drew, Johnson, Olesky, and van den Driessche.