Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 3

Abstract:

      Let $S$ be a nonempty, proper subset of all possible refined inertias of real matrices of order $n$. The set $S$ is a critical set of refined inertias for irreducible sign patterns of order $n$, if for each $n\times n$ irreducible sign pattern $\mathcal{A}$, the condition $S\subseteq ri(\mathcal{A})$ is sufficient for $\mathcal{A}$ to be refined inertially arbitrary. If no proper subset of $S$ is a critical set of refined inertias, then $S$ is a minimal critical set of refined inertias for irreducible sign patterns of order $n$. All minimal critical sets of refined inertias for full sign patterns of order $3$ have been identified in [Wei GAO, Zhongshan LI, Lihua ZHANG, The minimal critical sets of refined inertias for $3\times3$ full sign patterns, Linear Algebra Appl. 458(2014), 183--196]. In this paper, the minimal critical sets of refined inertias for irreducible sign patterns of order $3$ are identified.