Sign patterns that require $\mathbb{H}_n$ exist for each $n\geq 4$

TL;DR

This paper proves that for all $n \\geq 4$, there exist specific irreducible sign patterns of matrices that necessarily have refined inertia sets associated with Hopf bifurcation, countering previous conjectures.

Contribution

The authors identify three irreducible sign patterns for each $n \\geq 4$ that require the refined inertia set $\\mathbb{H}_n$, thus disproving a prior conjecture.

Findings

Existence of $n \\geq 4$ irreducible sign patterns requiring $\\mathbb{H}_n$

Counterexample to Bodine et al.'s conjecture for all $n \\geq 4$

Resolution of the conjecture about sign patterns and refined inertia sets

Abstract

The refined inertia of a square real matrix $A$ is the ordered $4$-tuple $(n_+, n_-, n_z, 2n_p)$, where $n_+$ (resp., $n_-$) is the number of eigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is the number of zero eigenvalues of $A$, and $2n_p$ is the number of nonzero pure imaginary eigenvalues of $A$. For $n \geq 3$, the set of refined inertias $\mathbb{H}_n=\{(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\}$ is important for the onset of Hopf bifurcation in dynamical systems. We say that an $n\times n$ sign pattern ${\cal A}$ requires $\mathbb{H}_n$ if $\mathbb{H}_n=\{\text{ri}(B) | B \in Q({\cal A})\}$. Bodine et al. conjectured that no $n\times n$ irreducible sign pattern that requires $\mathbb{H}_n$ exists for $n$ sufficiently large, possibly $n\ge 8$. However, for each $n \geq 4$, we identify three $n\times n$ irreducible sign patterns that require $\mathbb{H}_n$, which resolves this conjecture.