The matrix sign decomposition and its relation to the polar decomposition
type: Item
instance of: scholarly article
title: The matrix sign decomposition and its relation to the polar decomposition
zbMATH Open document ID: 0816.15012
DOI: 10.1016/0024-3795(94)90393-X
author: Nicholas J. Higham
published in: Linear Algebra and its Applications
publication date: 1995-07-20T00:00:00Z
review text: The sign function maps a matrix \(A\) with complex entries to a matrix \[ S = \text{sign} (A) = Z \left( \begin{matrix} - I & 0 \\ - 0 & I \end{matrix} \right) Z^{-1}, \] if \(A = ZJZ^{-1}\), where \(J\) is the Jordan form of \(A\) and \(J = \left( \begin{smallmatrix} J_ 1 & 0 \\ 0 & J_ 2 \end{smallmatrix} \right)\), where the eigenvalues of \(J_ 1\) lie in the open left half plane and those of \(J_ 2\) lie in the open right half plane. The matrix sign decomposition of \(A\) is defined by \(A = SN\). The author shows that there are several relationships and analogies between the matrix sign decomposition and the polar decomposition of a matrix. He obtains the formula \(S = A(A^ 2)^{-1/2}\), derives some perturbation result, establishes error bounds and applies his results to a family of iterations for computing \(S\).
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 720466
zbMATH Keywords: parallel computation
zbMATH Keywords: Jordan form
zbMATH Keywords: eigenvalues
zbMATH Keywords: matrix sign decomposition
zbMATH Keywords: polar decomposition
zbMATH Keywords: error bounds
Wikidata QID: Q56998974
reviewed by: Erich W. Ellers
describes a project that uses: Algorithm 674
describes a project that uses: CONEST
describes a project that uses: SONEST
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 720466