Sign patterns that require exist for each n ≥ 4

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• 作者：Wei Gao ; ^{wgao2@gsu.edu" class="auth_mail" title="E-mail the corresponding author} ; Zhongshan Li ; Lihua Zhang
• 关键词：15B35 ; 15A18 ; 05C50
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：489
• 期：Complete
• 页码：15-23
• 全文大小：263 K

文摘

The refined inertia of a square real matrix A   is the ordered 4-tuple (n₊,n₋,n_z,2n_p)$(n_{+},n_{-},n_z,2n_p)$, where n₊$n_{+}$ (resp., n₋$n_{-}$) is the number of eigenvalues of A   with positive (resp., negative) real part, n_z$n_z$ is the number of zero eigenvalues of A  , and 2n_p$2n_p$ is the number of nonzero pure imaginary eigenvalues of A  . For n≥3$n\ge 3$, the set of refined inertias H_n={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)}${\mathbb{H}}_n=\{(0,n,0,0),(0,n-2,0,2),(2,n-2,0,0)\}$ is important for the onset of Hopf bifurcation in dynamical systems. We say that an n×n$n\times n$ sign pattern A$\mathcal{A}$ requires H_n${\mathbb{H}}_n$ if H_n={ri(B)|B∈Q(A)}${\mathbb{H}}_n=\{\mathrm{ri}(B)|B\in Q(\mathcal{A})\}$. Bodine et al. conjectured that no n×n$n\times n$ irreducible sign pattern that requires H_n${\mathbb{H}}_n$ exists for n   sufficiently large, possibly n≥8$n\ge 8$. However, for each 17bf52f2" title="Click to view the MathML source">n≥4$n\ge 4$, we identify three n×n$n\times n$ irreducible sign patterns that require H_n${\mathbb{H}}_n$, which resolves this conjecture.