A continuous Jacobi-like approach to the simultaneous reduction of real matrices
type: Item
instance of: scholarly article
title: A continuous Jacobi-like approach to the simultaneous reduction of real matrices
zbMATH Open document ID: 0726.65044
DOI: 10.1016/0024-3795(91)90230-T
author: Moody T. Chu
published in: Linear Algebra and its Applications
publication date: 1991-01-01T00:00:00Z
review text: This paper approaches the simultaneous reduction of real and complex matrices in a novel way. Any number of matrices can be considered in this approach and nearly any desired simultaneous normal form can be specified. Specifically, the cases of orthogonal simultaneous diagonalization, that of the nearest commuting matrix pair, the orthogonal equivalence reduction and the question of the closest normal matrix are solved computationally. The QR, QZ and SVD algorithms and many others can be treated similarly. The method used models the problem of simultaneous matrix reduction by forming a differential equation and following its flow until optimally, the sum of the undesired portions of the achieved near normal matrices are minimized in a Jacobi like fashion. For each of the four treated specific problems, the paper develops a constrained optimization problem and gives several computed examples that are solved by using standard ordinary differential equation initial value problem software.
reviewed by: Frank Uhlig
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 65L05
zbMATH DE Number: 4199477
zbMATH Keywords: Jacobi method
zbMATH Keywords: similarity
zbMATH Keywords: optimization
zbMATH Keywords: gradient method
zbMATH Keywords: congruence transformation
zbMATH Keywords: orbit
zbMATH Keywords: QR algorithm
zbMATH Keywords: QZ algorithm
zbMATH Keywords: simultaneous reduction
zbMATH Keywords: normal form
zbMATH Keywords: orthogonal simultaneous diagonalization
zbMATH Keywords: nearest commuting matrix pair
zbMATH Keywords: orthogonal equivalence reduction
zbMATH Keywords: closest normal matrix
zbMATH Keywords: SVD algorithms
zbMATH Keywords: flow
zbMATH Keywords: ordinary differential equation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4199477