\(\phi_S\)-orthogonal matrices
type: Item
instance of: scholarly article
title: \(\phi_S\)-orthogonal matrices
zbMATH Open document ID: 1227.15026
author: Ma. Nerissa M. Abara
author: Dennis I. Merino
author: Agnes T. Paras
published in: Linear Algebra and its Applications
publication date: 2010-04-29T00:00:00Z
review text: For any \(S\in M_n(\mathbb{R})\) with \(S^2=I\) or \(S^2=-I\), define \(\phi_S(A)=S^{-1}A^TS\). A matrix \(A\in M_n(\mathbb{C})\) is called \(\phi_S\)-orthogonal if \(\phi_S(A)A=I\). In this paper, the authors show that any complex matrix \(A\) admits a polar decomposition \(A=PU\) such that \(P\) is positive definite, \(U\) is unitary and both factors are \(\phi_S\)-orthogonal. More detailed properties for \(\phi_S\)-orthogonal matrices are given when \(S\) is of the form \(H_k=\left(\begin{matrix} 0& I_k\\-I_k& 0\end{matrix}\right)\) or \(L_k=\left(\begin{matrix} I_k &0\\0& -I_{n-k}\end{matrix}\right)\).
reviewed by: Xiangqian Guo
Mathematics Subject Classification ID: 15B10
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 5702825
zbMATH Keywords: \(\phi_S\)-orthogonal matrix
zbMATH Keywords: \(\phi_S\)-skew symmetric matrix
zbMATH Keywords: polar decomposition
zbMATH Keywords: positive definite
zbMATH Keywords: unitary
MaRDI profile type: Publication
full work available at URL: j.laa.2009.12.029
OpenAlex ID: W2016214787
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DOI: 10.1016/J.LAA.2009.12.029
description: scientific article; zbMATH DE number 5702825