Tridiagonal normal forms for orthogonal similarity classes of symmetric matrices
type: Item
instance of: scholarly article
title: Tridiagonal normal forms for orthogonal similarity classes of symmetric matrices
zbMATH Open document ID: 1063.15008
author: Kaiming Zhao
author: Dragomir Ž. Đoković
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: Considering the orthogonal group of matrices of order \(n\) over an algebraically closed field of characteristic different from 2, this paper presents a study of similarity action of this group on the space of all symmetric matrices of order \(n\) over the mentioned field. It is shown that each group orbit in the space of all symmetric matrices has a representative that is the direct sum of tridiagonal blocks, whose super- and subdiagonal entries are 1, each block being similar to a single Jordan block. If the field has characteristic 0, then the mentioned tridiagonal blocks can be replaced with new tridiagonal blocks which can be (unique) symmetrized Jordan blocks, this allowing to obtain a new type of normal form for symmetric matrices. In this case, the known normal forms (according to the classical book of \textit{F. R. Gantmacher}, The theory of matrices (1953; Zbl 0050.24804)) are not tridiagonal. Finally, four open problems are formulated.
reviewed by: Mihail Voicu
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 2083493
zbMATH Keywords: orthogonal group
zbMATH Keywords: symmetric matrices
zbMATH Keywords: tridiagonal matrices
zbMATH Keywords: symmetrized Jordan blocks
zbMATH Keywords: normal forms
MaRDI profile type: Publication
full work available at URL: j.laa.2003.12.038
OpenAlex ID: W2170477486
cites work: Q4450706
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cites work: Matrix Analysis
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DOI: 10.1016/J.LAA.2003.12.038
description: scientific article; zbMATH DE number 2083493