On the class of \(D_{k}\)-symmetrizable matrices
type: Item
instance of: scholarly article
title: On the class of \(D_{k}\)-symmetrizable matrices
zbMATH Open document ID: 1032.15020
DOI: 10.1016/S0024-3795(01)00464-5
author: Sławomir Jenek
author: Tomasz Szulc
author: Frank Uhlig
published in: Linear Algebra and its Applications
publication date: 2003-05-04T00:00:00Z
review text: An \(n\times n\) real matrix \(X\) is said to be \(D_{k}\)-symmetrizable if there exists a nonsingular diagonal matrix \(D\) with \(s(D)=k\) and \(DX=X^{\prime }D\) , where \(X^{\prime }\) is the transpose of \(X\) and the signature \(s(D)\)\ is the absolute value of the difference of the number of positive eigenvalues and the number of negative eigenvalues. In this paper, the authors characterize the matrices \(D_{k}\)-symmetrizable proving that they have many properties similar to real symmetric matrices. They prove that if \(A\) is \(D_{k}\)-symmetrizable then, if \(k=n\), all the eigenvalues of \(A\) are real and \(s(A)=s(DA)\), and, if \(k<n\), \(A\) has at least \(k\) real eigenvalues. It is known that for every \(n\times n\) real matrix \(X\) there is a \(n\times n\) symmetric and nonsingular matrix \(S\) such that \(SX=X^{\prime }S\) with \(s(S)=k \), that is, \(X\) is \(S_{k}\)-symmetrizable. Then it is equivalent that \(A\) is \( S_{k}\)-symmetrizable and that there is an orthogonal \(Q\) such that \( Q^{\prime }AQ\) is \(D_{k}\)-symmetrizable. Moreover, they also prove that for any \(n\times n\) real matrix \(B\) there exist orthogonal \(P\) and \(Q\) such that \(PBQ\) is sign-symmetric (\( \text{sign}(x_{ij})= \text{sign}(x_{ji})\)) and \(PQ\) is a diagonal matrix with entries \(\pm 1 \), generalizing, in some sense, the singular value decomposition for real square matrices.
reviewed by: M. Ladra
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1903564
zbMATH Keywords: symmetric matrix
zbMATH Keywords: diagonal matrix
zbMATH Keywords: eigenvalues
zbMATH Keywords: orthogonal similarity
zbMATH Keywords: signature
zbMATH Keywords: symmetrizable matrix
zbMATH Keywords: singular value decomposition
MaRDI profile type: Publication
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full work available at URL: s0024-3795(01)00464-5
OpenAlex ID: W1990921965
description: scientific article; zbMATH DE number 1903564