Real tridiagonalization of Hermitian matrices by modified Householder transformation
type: Item
instance of: scholarly article
title: Real tridiagonalization of Hermitian matrices by modified Householder transformation
zbMATH Open document ID: 0861.65037
author: Osamu Shukuzawa
author: Toshio Suzuki
author: Ichiro Yokota
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-04-29T00:00:00Z
review text: Elementary Householder reflections are of the form \(I-\sigma ww^H\), where \(\sigma\in\mathbb{R}\) is even when \(w\in\mathbb{C}^n\). This short note considers extended reflections where \(\sigma\in\mathbb{C}\). It is shown that one can always transform any complex vector to a real multiple of the first unit vector using an extended reflection. As a consequence, Hermitian matrices can be reduced to real, symmetric tridiagonal form via two-sided transformations with a series of extended reflections. A similar result is shown to hold if the vector components are quaternions.
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 952367
zbMATH Keywords: real tridiagonalization
zbMATH Keywords: elementary matrices
zbMATH Keywords: complex numbers
zbMATH Keywords: Householder reflections
zbMATH Keywords: Hermitian matrices
zbMATH Keywords: real, symmetric tridiagonal form
zbMATH Keywords: quaternions
reviewed by: Andreas Frommer
MaRDI profile type: Publication
full work available at URL: pjaa.72.102
OpenAlex ID: W2044044184
cites work: Q3141892
cites work: Q4001758
cites work: Q5674306
cites work: Q3236366
cites work: On the cells of symplectic groups
DOI: 10.3792/PJAA.72.102
description: scientific article; zbMATH DE number 952367