On tridiagonal matrices unitarily equivalent to normal matrices
type: Item
instance of: scholarly article
title: On tridiagonal matrices unitarily equivalent to normal matrices
zbMATH Open document ID: 1221.15058
published in: Linear Algebra and its Applications
publication date: 2010-05-11T00:00:00Z
review text: Two matrices \(A\) and \(B\) of the same dimensions are equivalent if nonsingular matrices \(T\) and \(S\) exist such that \(A=S^{-1}BT\). Unitarily equivalent means that both \(S\) and \(T\) are unitary. In this paper, the unitary equivalence transformation of a matrix to tridiagonal form is studied. The main results are related to applying the tridiagonalization procedure to normal matrices. The resulting tridiagonal matrix yields interesting properties related to its super- and subdiagonal elements. It is shown that these super- and subdiagonal elements have the same absolute value. A Householder tridiagonalization scheme as well as an iterative method and its relation to Krylov subspaces are presented. Several possibilities for reducing matrices are discussed and are illustrated by well-known classes of normal matrices. Specific results are obtained when the algorithm is applied to symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is also proved. Finally, it is shown that the reduction can be used to compute the singular value decomposition of normal matrices making use of Takagi factorization. Some extra properties of the tridiagonalization procedure are presented together with the efficient method for computing a unitary complex symmetric decomposition of a normal matrix.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F20
zbMATH DE Number: 5706968
zbMATH Keywords: normal matrices
zbMATH Keywords: complex symmetric matrices
zbMATH Keywords: Takagi factorization
zbMATH Keywords: tridiagonal matrices
zbMATH Keywords: singular values
zbMATH Keywords: unitary equivalence
zbMATH Keywords: unitary-complex symmetric factorization
zbMATH Keywords: Krylov subspaces
zbMATH Keywords: tridiagonalization
zbMATH Keywords: algorithm
zbMATH Keywords: singular value decomposition
author: Raf Vandebril
MaRDI profile type: Publication
full work available at URL: j.laa.2010.02.009
OpenAlex ID: W2067378250
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DOI: 10.1016/J.LAA.2010.02.009
description: scientific article; zbMATH DE number 5706968