Singular-value-like decomposition for complex matrix triples
type: Item
instance of: scholarly article
title: Singular-value-like decomposition for complex matrix triples
zbMATH Open document ID: 1186.65048
author: Christian Mehl
author: Volker Mehrmann
author: Hongguo Xu
published in: Journal of Computational and Applied Mathematics
publication date: 2009-12-15T00:00:00Z
review text: Instead of computing the singular values of a general matrix \(A \in \mathbb C^{m \times n}\) and thus revealing the eigenvalues of \(AA^*\) and \(A^*A\), one may ask for a canonical form for \(A\) that reveals the eigenvalues of the complex symmetric matrices \(AA^T\) and \(A^TA\). The authors obtain such a form by solving a more general problem considering a triple of matrices \((A,G, \hat G)\) with \(A \in \mathbb C^{m \times n}, G \in \mathbb C^{m \times m}\) and \(\hat G \in \mathbb C^{n \times n}\), where \(G\) and \(\hat G\) are nonsingular and either complex symmetric or complex skew-symmetric. Then they derive canonical forms for matrix triples \((A, G, \hat G)\) that provide the Jordan canonical forms of the structured matrices \(\hat {\mathcal H} = \hat G^{-1}A^TG^{-1}A\) and \({\mathcal H} = G^{-1}A\hat G^{-1}A^T\).
reviewed by: Petko Hr. Petkov
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5647946
zbMATH Keywords: singular value decomposition
zbMATH Keywords: canonical form
zbMATH Keywords: Hamiltonian matrix
zbMATH Keywords: complex bilinear form
zbMATH Keywords: complex symmetric matrix
zbMATH Keywords: complex skew-symmetric matrix
zbMATH Keywords: Takagi factorization
MaRDI profile type: Publication
full work available at URL: j.cam.2008.02.017
OpenAlex ID: W2091735253
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DOI: 10.1016/J.CAM.2008.02.017
description: scientific article; zbMATH DE number 5647946