Some results on a partition of \(Z\)-matrices
type: Item
instance of: scholarly article
title: Some results on a partition of \(Z\)-matrices
zbMATH Open document ID: 0829.15011
DOI: 10.1016/0024-3795(94)00168-D
author: Ronald L. Smith
published in: Linear Algebra and its Applications
publication date: 1996-01-24T00:00:00Z
review text: A \(Z\)-matrix is a real matrix in which each off-diagonal entry is nonpositive. For each nonnegative \(n \times n\) matrix \(B\) and \(k = 1, \ldots, n\), we denote by \(\rho_k (B)\) the maximum of the spectral radii of the principal \(k \times k\) submatrices of \(B\); and we put \(\rho_0 (B) = - \infty\) and \(\rho_{n + 1} (B) = \infty\). \textit{M. Fiedler} and \textit{T. L. Markham} [Linear Algebra Appl. 173, 115-124 (1992; Zbl 0754.15016)] considered the classes \(L_k\) \((k = 0, \ldots, n)\) consisting of real \(n \times n\) matrices of the form \(tI - B\) with \(B\) nonnegative and \(\rho_k (B) \leq t < \rho_{k + 1} (B)\). They showed that every \(n \times n\) \(Z\)-matrix lies in exactly one of these classes. The present paper considers properties of \(L_k\)-matrices and their Schur complements. A typical result is as follows (Theorem 2.1): An \(n \times n\) \(Z\)-matrix \(A\) is an \(L_k\)-matrix if and only if \(A^{-1}\) has a principal minor of order \(n - k - 1\) which is opposite in sign to \(\text{det} A\), but no principal minor of \(A^{-1}\) of larger order has this property.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 778277
zbMATH Keywords: partition
zbMATH Keywords: spectral radius
zbMATH Keywords: \(Z\)-matrix
zbMATH Keywords: Schur complements
MaRDI profile type: Publication
cites work: Q3943082
cites work: Notes on \(F_ 0\)-matrices
cites work: An Identity for the Schur Complement of a Matrix
cites work: A classification of matrices of class Z
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cites work: A generalization of N-matrices
cites work: Nonnegative Matrices whose Inverses are M-Matrices
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cites work: The Schur complement of a generalized M-matrix
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description: scientific article; zbMATH DE number 778277