Full sign-invertibility and symplectic matrices
type: Item
instance of: scholarly article
title: Full sign-invertibility and symplectic matrices
zbMATH Open document ID: 0838.05016
DOI: 10.1016/0024-3795(94)00032-8
author: David A. Schmidt
author: Chjan C. Lim
published in: Linear Algebra and its Applications
publication date: 1996-05-30T00:00:00Z
full work available at URL: http://purl.umn.edu/2574
review text: From authors' abstract: ``An \(n\times n\) sign pattern \(H\) is said to be sign-invertible if there exists a sign pattern \(H^{-1}\) (called the sign inverse of \(H\)) such that, for all matrices \(A\in Q(H)\) (the sign class of \(H\)), \(A^{- 1}\) exists and \(A^{-1}\in Q(H^{- 1})\). If, in addition, \(H^{-1}\) is sign-invertible, \(H\) is said to be fully sign-invertible and \((H, H^{- 1})\) is called a sign-invertible pair. Given an \(n\times n\) sign pattern \(H\), a symplectic pair in \(Q(H)\) is a pair of matrices \((A, D)\) such that \(A\in Q(H)\), \(D\in Q(H)\), and \(A^TD= I\).'' The authors discuss the digraphical relationship between a sign-invertible pattern \(H\) and its sign inverse \(H^{ -1}\), and use this to cast a necessary condition for full sign-invertibility of \(H\). They proceed to develop sufficient conditions for full sign-invertibility of \(H\) in terms of allowed paths and cycles in the digraph of \(H\), and conclude with a complete characterization of those sign patterns that require symplectic pairs.
reviewed by: K.Burian
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 841756
zbMATH Keywords: symplectic matrices
zbMATH Keywords: sign pattern
zbMATH Keywords: sign-invertible
zbMATH Keywords: symplectic pair
zbMATH Keywords: sign inverse
zbMATH Keywords: sign-invertibility
zbMATH Keywords: paths
zbMATH Keywords: cycles
zbMATH Keywords: digraph
MaRDI profile type: Publication
OpenAlex ID: W2016341750
cites work: Binary trees, symplectic matrices and the Jacobi coordinates of celestial mechanics
cites work: Q3974962
cites work: Qualitative Economics and the Scope of the Correspondence Principle
cites work: Qualitatively invertible matrices
cites work: When the sign pattern of a square matrix determines uniquely the sign pattern of its inverse
cites work: Nonsingular sign patterns and the orthogonal group
description: scientific article; zbMATH DE number 841756