Some notes on Z-matrices
type: Item
instance of: scholarly article
title: Some notes on Z-matrices
zbMATH Open document ID: 0652.15017
DOI: 10.1016/0024-3795(88)90030-4
author: Ronald L. Smith
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: This paper deals with new results on Z- and M-matrices. First, a irreducibility criterion is proved: For a matrix being partitioned into four blocks with one being nonsingular in the main diagonal and those being in the second diagonal being nonzero, if the Schur complement of the matrix being nonsingular is irreducible, then the overall matrix is irreducible. Furthermore, if the overall matrix is an irreducible Z-matrix and the nonsingular block matrix is an M-matrix then the Schur complement of such a block matrix is also irreducible. An inequality is obtained relating the eigenvalue of minimum modulus of an \(N_ 0\)-matrix to the eigenvalue of minimum modulus of the Schur complement in that matrix of an M- submatrix. Similar results are obtained for irreducible M-matrices and irreducible nonsingular \(F_ 0\)-matrices. Circulant \(N_ 0\)-matrices of order three are characterized in terms of their spectrum while matrices of \(F_ 0\) inverses are shown to have nonpositive principal minors of order \(\geq 2\).
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 4063282
zbMATH Keywords: irreducible matrices
zbMATH Keywords: Z-matrices
zbMATH Keywords: M-matrices
zbMATH Keywords: Schur complement
zbMATH Keywords: block matrix
zbMATH Keywords: \(N_ 0\)-matrix
zbMATH Keywords: eigenvalue of minimum modulus
zbMATH Keywords: \(F_ 0\)-matrices
reviewed by: M. De la Sen
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90030-4
OpenAlex ID: W2067749639
cites work: Generalized Inverse Formulas Using the Schur Complement
cites work: An Identity for the Schur Complement of a Matrix
cites work: Some matrix inequalities
cites work: Classes of products of M-matrices and inverse M-matrices
cites work: Q5340130
cites work: A generalization of N-matrices
cites work: Inverse \(N_ 0\)-matrices
cites work: Nonnegative Matrices whose Inverses are M-Matrices
cites work: The Schur complement of a generalized M-matrix
description: scientific article; zbMATH DE number 4063282