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).
 Author Name Affiliation Yajing WANG Department of Data Science and Technology, North University of China, Shanxi 030051, P. R. China Yubin GAO Department of Mathematics, North University of China, Shanxi 030051, P. R. China Yanling SHAO Department of Mathematics, North University of China, Shanxi 030051, P. R. China
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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.