The \(\phi_S\) polar decomposition
type: Item
instance of: scholarly article
title: The \(\phi_S\) polar decomposition
zbMATH Open document ID: 1262.15011
author: Daryl Q. Granario
author: Dennis I. Merino
author: Agnes T. Paras
published in: Linear Algebra and its Applications
publication date: 2013-01-21T00:00:00Z
review text: Let \(S_n^+ = \{S\in M_n(\mathbb{R})\mid S^2=1\}\), \(S_n^- = \{S\in M_n(\mathbb{R})\mid S^2=-1\}\), and \(S_n = S_n^+\cup S_n^-\). For \(S\in S_n\) let \(\phi_S : M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})\); \(A\rightarrow S^{-1}A^TS\). With respect to \(\phi_S\), an element \(A\in M_n(\mathbb{C})\) is called symmetric if \(\phi_S (A) = A\), skew symmetric if \(\phi_S (A) = - A\), and orthogonal if \(A\) is nonsingular and \(\phi_S (A) = A^{-1}\). If \(A = QT\), where \(Q\) is orthogonal and \(T\) is symmetric, then \(A\) is said to have a \(\phi_S\) polar decomposition. The authors show that every nonsingular \(A\in M_n(\mathbb{C})\) such that \(\phi_S(\phi_S(A)) = A\) has a \(\phi_S\) polar decomposition. They also discuss possible Jordan canonical forms.
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15B10
zbMATH DE Number: 6127365
zbMATH Keywords: polar decomposition
zbMATH Keywords: symmetric matrix
zbMATH Keywords: orthogonal matrix
zbMATH Keywords: skew symmetric matrix
zbMATH Keywords: Jordan canonical forms
MaRDI profile type: Publication
full work available at URL: j.laa.2012.07.026
OpenAlex ID: W1983390144
cites work: \(\phi_S\)-orthogonal matrices
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cites work: Matrix Analysis
cites work: Q4040931
cites work: Contragredient equivalence: A canonical form and some applications
cites work: The Jordan canonical forms of complex orthogonal and skew-symmetric matrices
cites work: Algebraic Polar Decomposition
DOI: 10.1016/J.LAA.2012.07.026
description: scientific article; zbMATH DE number 6127365