\(\mathbb G\)-reflectors: Analogues of Householder transformations in scalar product spaces
type: Item
instance of: scholarly article
title: \(\mathbb G\)-reflectors: Analogues of Householder transformations in scalar product spaces
zbMATH Open document ID: 1059.65040
author: D. Steven Mackey
author: Niloufer Mackey
author: Françoise Tisseur
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: An elementary reflector (also called Householder transformation) is a unitary matrix of the form \(H = I-2vv^\star\), where \(I\) is the identity matrix and \(v\) is some vector with \(\| v\| _2=1\). For any two vectors \(p\) and \(q\) with \(\| p\| _2 = \| q\| _2\) there exists such a transformation which maps \(p\) to \(q\). This mapping property makes elementary reflectors an important tool for developing algorithms in numerical linear algebra. Structure-preserving algorithms often rely on variants of elementary reflectors which yield, e.g., complex orthogonal, symplectic or pseudo-orthogonal transformations. This paper provides a general and comprehensive framework for such variants. If \(\langle \cdot, \cdot\rangle\) is a bilinear form (\(\langle \cdot, \cdot\rangle = x^T My\) for some matrix \(M\)) or a sesquilinear form (\(\langle \cdot, \cdot\rangle = x^\star My\) for some matrix \(M\)) then the automorphism group \(\mathbb G\) associated with \(\langle \cdot, \cdot\rangle\) consists of all matrices \(G\) satisfying \(\langle G x, Gy\rangle = \langle x, y\rangle\). A \(\mathbb G\)-reflector is defined as a transformation \(G = I + uv^\star\) with \(G \in \mathbb G\). It is shown that such a \(G\) can always be written in the form \(G = I + \beta uu^T\) (bilinear \(\langle \cdot, \cdot\rangle\)) or \(G = I + \beta uu^\star\) (sesquilinear \(\langle \cdot, \cdot\rangle\)), provided that \(M\) is nonsingular. The usefulness of \(\mathbb G\)-reflectors for dealing with structured matrices stems from the fact that the transformation \(A\rightarrow G^{-1} A G\) not only preserves the automorphism group but also the Jordan and Lie algebras associated with \(\langle \cdot, \cdot\rangle\). For example, if \(\langle x,y \rangle = x^\star y\) then \(\mathbb G\) consists of unitary matrices and the transformation above preserves Hermitian as well as skew-Hermitian matrices. This paper gives geometric interpretations of \(G\) depending on the question whether \(u\) is is…
reviewed by: Daniel Kressner
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 2083515
zbMATH Keywords: scalar product
zbMATH Keywords: bilinear form
zbMATH Keywords: sesquilinear form
zbMATH Keywords: Householder transformations
zbMATH Keywords: structure-preserving algorithm
zbMATH Keywords: reflector
zbMATH Keywords: algorithm
zbMATH Keywords: orthosymmetric
zbMATH Keywords: isotropic
zbMATH Keywords: hyperbolic transformation
zbMATH Keywords: symmetries
zbMATH Keywords: transvections
zbMATH Keywords: symplectic
zbMATH Keywords: pseudo-unitary
describes a project that uses: CLARFG
describes a project that uses: NAG
describes a project that uses: LAPACK
describes a project that uses: nag
describes a project that uses: mctoolbox
MaRDI profile type: Publication
full work available at URL: j.laa.2003.07.009
OpenAlex ID: W2110773940
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DOI: 10.1016/J.LAA.2003.07.009
description: scientific article; zbMATH DE number 2083515