Stability of self-adjoint square roots and polar decompositions in indefinite scalar product spaces
type: Item
instance of: scholarly article
title: Stability of self-adjoint square roots and polar decompositions in indefinite scalar product spaces
zbMATH Open document ID: 0954.15008
DOI: 10.1016/S0024-3795(99)00072-5
author: André C. M. Ran
author: Leiba Rodman
author: C. van der Mee
published in: Linear Algebra and its Applications
publication date: 2000-06-07T00:00:00Z
review text: Let \(F\) be either the real or complex numbers and let \(H\) be an invertible matrix that is either real symmetric or complex Hermitian as appropriate. The usual scalar product \(\langle x,y\rangle\) on \(F^n\) can then be modified by \(H\) to give a perhaps indefinite scalar product \([x,y]=\langle Hx,y \rangle\). The \(H\)-adjoint \(A^H\) of an \(n\times n\) matrix \(A\) is defined by \([Ax, y]= [x,A^Hy]\) and an \(n\times n\) matrix \(U\) is \(H\)-unitary if \([Ux,Uy] =[x,y]\). An \(H\)-polar decomposition of a square matrix \(X\) is one of the form \(X=UA\) with \(U\) \(H\)-polar and \(A\) \(H\)-self-adjoint. The authors are concerned with the existence of stable \(H\)-polar decompositions, that is, decompositions that depend continuously on \(X\) for small perturbations, when the scalar product is genuinely indefinite. In the complex case, such a decomposition occurs if and only if the eigenvalues of \(X^HX\) are positive. In the real case, a further requirement is needed for necessity. The authors also consider related questions concerning matrices that can be written as \(X^HX\), and those matrices \(Y\) having an \(H\)-self-adjoint square root. A characterization is given of such \(Y\) whose square root depends continuously on \(Y\).
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 1419861
zbMATH Keywords: indefinite scalar product
zbMATH Keywords: square root of matrix
zbMATH Keywords: polar decomposition
zbMATH Keywords: adjoint matrix
zbMATH Keywords: unitary matrix
MaRDI profile type: Publication
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Wikidata QID: Q127723541
description: scientific article; zbMATH DE number 1419861