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Zero-nonzero patterns that allow or require an inertia set related to dynamical systems. (English) Zbl 1452.15019

Authors’ abstract: The inertia of an \(n\times n\) real matrix \(B\), denoted by \(i(B)\), is the ordered triple \(i(B) = (i_+(B), i_{-}(B); i_0(B)),\) in which \(i_+(B), i_-(B)\) and \(i_0(B)\) are the numbers of its eigenvalues (counting multiplicities) with positive, negative and zero real parts, respectively. The inertia of an \(n\times n\) zero-nonzero pattern \(A\) is the set \(i(A) = \{i(B) \mid B \in Q(A)\}\). For \(n\ge 2\), let \(S_n^* = \{(0, n, 0), (0, n -1,1), (1, n - 1, 0), (n, 0, 0), (n - 1, 0, 1), (n -1, 1, 0)\}\). An \(n\times n\) zero-nonzero pattern \(A\) allows \(S_n^*\) if \(S_n^*\subseteq i(A)\) and requires \(S_n^*= i(A)\). In this paper, it is shown that there are no zero-nonzero patterns for order \(n\ge 2\) that require \(S_n^*\). Also, a complete characterization of zero-nonzero star patterns of order \(n\ge 3\) that allow \(S_n^*\) is given.

MSC:

15B36 Matrices of integers
15B35 Sign pattern matrices
05C20 Directed graphs (digraphs), tournaments
15A18 Eigenvalues, singular values, and eigenvectors
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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