Nested sequences of principal minors and potential stability
type: Item
instance of: scholarly article
title: Nested sequences of principal minors and potential stability
zbMATH Open document ID: 0880.15025
DOI: 10.1016/S0024-3795(97)80034-1
author: Charles R. Johnson
author: John S. Maybee
author: D. D. Olesky
author: P. van den Driessche
published in: Linear Algebra and its Applications
publication date: 1997-08-26T00:00:00Z
review text: An \(n\times n\) sign-pattern \(A=[a_{ij}]\) with \(a_{ij}\in \{+,0,- \}\) defines a sign-pattern class \(Q(A)\), consisting of all real \(n\times n\) matrices \(B=[b_{ij}]\) with \(\text{sign } (b_{ij})=a_{ij}\). A sign-pattern \(A\) is said to be stable if all matrices \(B\in Q(A)\) are stable (i.e. the eigenvalues of \(B\) have negative real parts), and potentially stable if there exists a stable \(B\in Q(A)\). If \((\alpha_1,\dots,\alpha_n)\) is a permutation of \((1,\dots,n)\) let \(B(\alpha_1,\dots,\alpha_k)\) be the submatrix of \(B\), corresponding to rows and columns \(\alpha_1,\dots,\alpha_k\). A sign-pattern \(A\) allows a nested sequence of properly signed principal minors if there exist \(B\in Q(A)\) and \((\alpha_1,\dots,\alpha_n)\) such that \(\text{sign(det} (B(\alpha_1,\dots,\alpha_k)))= (-1)^k\) for \(k=1,\dots,n\). In this paper the authors give sufficient conditions for a sign-pattern \(A\) to allow a nested sequence of non-zero principal minors. A method is given to \(\text{sign }A\) so that it allows a nested sequence of properly signed principal minors. In this case, according to a classical result of \textit{M. E. Fisher} and \textit{A. T. Fuller} [Proc. Cambridge Philos. Soc. 54, 417-425 (1958; Zbl 0085.33102)] and \textit{C. S. Ballantine} [Proc. Am. Math. Soc. 25, 728-734 (1970; Zbl 0228.15001)] the sign-pattern \(A\) is potentially stable. Necessary and sufficient conditions for a certain tree sign-pattern to be potentially stable are also given.
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1040644
zbMATH Keywords: nested sequences of principal minors
zbMATH Keywords: potential stability
zbMATH Keywords: stability of sign-pattern
zbMATH Keywords: sign-pattern
zbMATH Keywords: non-zero principal minors
reviewed by: Mihail M. Konstantinov
MaRDI profile type: Publication
cites work: Stabilization by a Diagonal Matrix
cites work: Positive feedback may sometimes promote stability
cites work: Q4858123
cites work: Q3256523
cites work: Matrix Analysis
cites work: Some Sign Patterns that Preclude Matrix Stability
cites work: Qualitative stability of linear systems
cites work: The potentially stable tree sign patterns for dimensions less than five
full work available at URL: s0024-3795(97)80034-1
OpenAlex ID: W1968741513
Recommended article: Sign patterns with a nest of positive principal minors
Recommended article: Some Sign Patterns that Preclude Matrix Stability
Recommended article: The Distance of Potentially Stable Sign Patterns to the Unstable Matrices
Recommended article: Q5153529
Recommended article: Constructions for potentially stable sign patterns
Recommended article: The sepr-sets of sign patterns
Recommended article: Allow problems concerning spectral properties of sign pattern matrices: a survey
Recommended article: The potentially stable tree sign patterns for dimensions less than five
Recommended article: Sign patterns that require repeated eigenvalues
Recommended article: A combinatorial converse to the Perron-Frobenius theorem
description: scientific article; zbMATH DE number 1040644