A sort-Jacobi algorithm for semisimple Lie algebras
type: Item
instance of: scholarly article
title: A sort-Jacobi algorithm for semisimple Lie algebras
zbMATH Open document ID: 1177.65056
author: Martin Kleinsteuber
published in: Linear Algebra and its Applications
publication date: 2008-12-02T00:00:00Z
review text: The classical Jacobi algorithm is extended to an unified Lie algebraic approach. The conventional Jacobi algorithm minimizes the distance to diagonality; they reduce the off-norm, i. e. the sum of squares of off-diagonal entries. Sorting the diagonal elements after each step would accelerate the convergence but, there are difficulties to apply this sorting to the off-norm, that has to minimize. Using the gradient flow of a trace function [\textit{R. W. Brockett}, Linear Algebra Appl. 146, 79--91 (1991; Zbl 0719.90045)] as a more appropriate distance measure matrices can be diagonalized, and the eigenvalues can be simultaneously sorted. In this paper the sort-Jacobi algorithm is extended to a large class of structured matrices, besides the semisimple Lie algebra cases, such as e. g. the symmetric, Hermitian, and skew-symmetric eigenvalue problem, and the real and complex singular value decomposition. The local quadratic convergence of the sort-Jacobi method is proved for the regular case but not for the case in which the eigenvalues or singular values occur in clusters. For these more complicated irregular problem a future publication is announced. The results are illustrated considering the Lie algebra of derivations of the complex octonions.
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 65F20
zbMATH DE Number: 5376893
zbMATH Keywords: sort-Jacobi algorithm
zbMATH Keywords: Cartan decomposition
zbMATH Keywords: restricted-root space
zbMATH Keywords: off-norm
zbMATH Keywords: trace function
zbMATH Keywords: structured eigenvalue decomposition
zbMATH Keywords: structured singular value decomposition
zbMATH Keywords: local quadratic convergence
zbMATH Keywords: exceptional Lie algebra
zbMATH Keywords: convergence acceleration
MaRDI profile type: Publication
full work available at URL: j.laa.2008.07.004
OpenAlex ID: W1985432759
Wikidata QID: Q115344881
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DOI: 10.1016/J.LAA.2008.07.004
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description: scientific article; zbMATH DE number 5376893