Some properties of GM-matrices and their inverses
type: Item
instance of: scholarly article
title: Some properties of GM-matrices and their inverses
zbMATH Open document ID: 0659.15021
DOI: 10.1016/0024-3795(88)90247-9
author: John S. Maybee
author: Daniel J. Richman
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: This article studies the relation between the cycle structure of the signed digraph of a matrix and the signs of its principal minors. An \(n\times n\) matrix A is a GM-matrix if its diagonal elements are negative and each negative cycle is contained in any positive cycle it intersects. (The sign of a cycle is the product of the signs of its arcs.) The matrix is consistently signed if each \(p\times p\) principal minor has sign \((- 1)^ p.\) The authors classify irreducible GM-matrices as follows: I. All cycles are negative (L-matrix). II. All cycles are positive (simply related to Z-matrices and M-matrices). III. There are positive and negative cycles, and every positive cycle has length n (degenerate case). IV. There are positive and negative cycles, and some positive cycle has length less than n. Conditions are obtained for a matrix of class III or IV to be consistently signed, and a canonical form is specified for class IV. For consistently signed class IV matrices, some information about elements of the inverse is found.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 4077451
zbMATH Keywords: qualitative matrix
zbMATH Keywords: stability
zbMATH Keywords: structure
zbMATH Keywords: signed digraph
zbMATH Keywords: principal minors
zbMATH Keywords: GM-matrix
zbMATH Keywords: cycle
zbMATH Keywords: irreducible
zbMATH Keywords: L-matrix
zbMATH Keywords: Z-matrices
zbMATH Keywords: M-matrices
zbMATH Keywords: degenerate
zbMATH Keywords: consistently signed
zbMATH Keywords: inverse
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90247-9
OpenAlex ID: W1990038414
cites work: Q4326384
cites work: Rectangular Matrices and Signed Graphs
cites work: Qualitatively invertible matrices
description: scientific article; zbMATH DE number 4077451