Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 3 
Received:January 25, 2019 Revised:December 08, 2019 
Key Word:
sign pattern refined inertia refined inertially arbitrary sign pattern critical set of refined inertias

Fund ProjectL:Supported by Shanxi Province Science Foundation for Youths (Grant No.201901D211227). 

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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), 183196]. In this paper, the minimal critical sets of refined inertias for irreducible sign patterns of order $3$ are identified. 
Citation: 
DOI:10.3770/j.issn:20952651.2020.03.002 
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