On nonsingular sign regular matrices
type: Item
instance of: scholarly article
title: On nonsingular sign regular matrices
zbMATH Open document ID: 1025.15038
DOI: 10.1016/S0024-3795(02)00437-8
author: J. M. Peña
published in: Linear Algebra and its Applications
publication date: 2003-03-23T00:00:00Z
review text: An \(m\times n\) real matrix \(A\) is called sign regular of order \(k\), \(1\leq k\leq \min\{m,n\}\) if for each \(l\), \(l=1,\ldots,k\), all \(l\times l\)-minors of \(A\) have the same sign \(\varepsilon_l\). The matrix \(A\) is called sign regular if \(k = \min\{m,n\}\). The list \((\varepsilon_1, \ldots,\varepsilon_k)\) of signs is called the signature of \(A\). A matrix is totally nonpositive (nonnegative) if all its minors are nonpositive (nonnegative). The paper deals with the zero pattern of nonsingular matrices which are sign regular of order \(k\). The author shows that the amount of zero entries that can appear in a nonsingular sign regular matrix depends on its signature. A characterization of nonsingular totally nonpositive matrices is obtained and a test of \(O(n^3)\) operations to check, if a square matrix of size \(n\) is nonsingular totally nonpositive, is provided. A key tool for the test is a Neville elimination procedure. Nonsingular matrices with all principle minors being nonpositive (called weak \(N\)-matrices) are considered. It is known that all principal minors of a nonsingular totally nonnegative matrix are positive. However, a similar result does not hold for a nonsingular totally nonpositive matrix. It is proved that if \(A=(a_{ij})\) is a nonsingular totally nonpositive matrix with \(a_{11}<0\) and \(a_{nn}<0\) then \(A\) is a weak \(N\)-matrix.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1886829
zbMATH Keywords: sign regular matrices
zbMATH Keywords: zero pattern
zbMATH Keywords: totally nonpositive matrices
zbMATH Keywords: \(N\)-matrices
zbMATH Keywords: Neville elimination
reviewed by: Alexander Guterman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1886829