Decompositions of strictly sign regular matrices
type: Item
instance of: scholarly article
title: Decompositions of strictly sign regular matrices
zbMATH Open document ID: 1148.15009
published in: Linear Algebra and its Applications
publication date: 2008-08-06T00:00:00Z
review text: An \(n\times n\) matrix \(A\) is strictly sign regular (SSR) if all its minors have the same strict sign. A matrix is totally positive (TP) if all its minors are positive. A triangular matrix \(A\) is called \(\Delta\)TP if its nontrivial minors are all positive. A nonsingular matrix \(A\) is said to be lowerly signed with signature \(\varepsilon_1,\dots,\varepsilon_n\) if it can be decomposed in the form \(A = LDU\) and \(LD\Sigma\) is \(\Delta\)TP, where \(\Sigma\) is a diagonal matrix with diagonal entries \(\varepsilon_1, \varepsilon_1\varepsilon_2 , \dots , \varepsilon_{n-1}\varepsilon_n\). If, in addition, \(U^{-1}\) satisfies that \(\Sigma U^{-1}\Sigma\) is \(\Delta\)TP, then the matrix is called strictly signed \(\gamma\)-matrix with signature \(\varepsilon_1 ,\dots , \varepsilon_n\). The authors study several decompositions of strictly sign regular matrices. They give the following characterization of strictly sign regular matrices: Let \(A\) be an \(n \times n\) matrix. Then \(A\) is SSR with signature \(\varepsilon_1 ,\dots , \varepsilon_n\) if and only if for each \(i = 1, \dots , n\), \[ A[i, \dots , n] = Q_i R_i ,\quad (A[i,\dots , n])^T = \widetilde Q_i\widetilde R_i , \] where \(Q_i\), \(\widetilde Q_i\) are orthogonal strictly signed \(\gamma\)-matrices with signature \(\varepsilon_1 ,\dots , \varepsilon_{n-i+1}\), and \(R_i\), \(\widetilde R_i\) are nonsingular and upper triangular \(\Delta\)TP matrices. They get similar results related to Schur decomposition, singular value decomposition, and symmetric-triangular decomposition.
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 5307050
zbMATH Keywords: strictly sign regular matrix
zbMATH Keywords: QR decomposition
zbMATH Keywords: Schur decomposition
zbMATH Keywords: symmetric-triangular decomposition
zbMATH Keywords: totally positive matrix
zbMATH Keywords: singular value decomposition
author: Vanesa Cortés
author: J. M. Peña
MaRDI profile type: Publication
full work available at URL: j.laa.2007.05.014
OpenAlex ID: W2074201869
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DOI: 10.1016/J.LAA.2007.05.014
description: scientific article; zbMATH DE number 5307050