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

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.