3阶不可约符号模式的精细惯量极小临界集

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

中文关键词:  符号模式  精细惯量  精细惯量任意的符号模式  精细惯量临界集

英文关键词:sign pattern  refined inertia  refined inertially arbitrary sign pattern  critical set of refined inertias

中文摘要:

      令$S$为$n$阶实矩阵所蕴含的所有精细惯量构成的集合的一个非空真子集.如果对于每一个$n\times n$不可约符号模式$A$, $S\subseteq ni(A)$是$A$是精细惯量任意的充分条件,则称集合$S$ 为$n$阶不可约符号模式的一个精细惯量临界集.如果$S$的任意一个真子集都不是一个精细惯量临界集,则$S$为$n$阶不可约符号模式的一个精细惯量极小临界集. 文献[Wei GAO, Zhongshan LI, Lihua ZHANG,The minimal critical sets of refined inertias for full sign patterns, Linear Algebra Appl. 458(2014), 183--196]中定义了3阶全符号模式的所有的精细惯量极小临界集.本文中,我们给出了3阶不可约符号模式的所有的精细惯量极小临界集.

英文摘要:

      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.