Zero-nonzero Patterns that Allow or Require an Inertia Set Related to Dynamical Systems

Abstract

The inertia of an $n\times n$ real matrix $B$, denoted by $\hbox{i}(B)$, is the ordered triple $\hbox{i}(B)=(i_+(B)$, $i_-(B), i_0(B) )$, in which $i_+(B)$, $i_-(B)$ and $i_0(B)$ are the numbers of its eigenvalues (counting multiplicities) with positive, negative and zero real parts, respectively. The inertia of an $n\times n$ zero-nonzero pattern ${\cal A}$ is the set $\hbox{i}({\cal A})=\{\hbox{i}(B) \mid B\in Q({\cal A})\}$. For $n\ge 2$, let $$\mathbb{S}_n^*=\{(0,n,0), (0,n-1,1), (1,n-1,0), (n,0,0), (n-1,0,1), (n-1,1,0)\}.$$

An $n\times n$ zero-nonzero pattern ${\cal A}$ allows $\mathbb{S}_n^*$ if $\mathbb{S}_n^* \subseteq \hbox{i}({\cal A})$ and requires $\mathbb{S}_n^*$ if $\mathbb{S}_n^*=\hbox{i}({\cal A})$. In this paper, it is shown that there are no zero-nonzero patterns for order $n\ge 2$ that require $\mathbb{S}_n^*$. Also, a complete characterization of zero-nonzero star patterns of order $n\ge 3$ that allow $\mathbb{S}_n^*$ is given.