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

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 $n\ge 3$, 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 $\mathcal{A}$ requires ${\mathbb{H}}_n$ if ${\mathbb{H}}_n=\{\text{ri}(B)|B\in Q(\mathcal{A})\}$. 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 $n\ge 4$, we identify three $n\times n$ irreducible sign patterns that require ${\mathbb{H}}_n$, which resolves this conjecture.