Normal forms in matrix spaces
type: Item
instance of: scholarly article
title: Normal forms in matrix spaces
zbMATH Open document ID: 0971.65037
DOI: 10.1007/BF01291714
published in: Integral Equations and Operator Theory
publication date: 2001-10-25T00:00:00Z
review text: The action of the group \(G_{lr}[m,n] \equiv GL(m,\mathbb{C}) \times GL(n,\mathbb{C})\) on the space of \(m \times n\) matrices over \(\mathbb{C}\) is classified and certain class of subgroups of \(G_{lr}[m,n]\) which are called admissible are introduced. An algorithm that reduces any matrix to a normal form with respect to the action of an admissible group is given. The algorithm is based on the fact that the action of an admissible group corresponds to a partition of an \(m \times n\) matrix into blocks, each block corresponding to a pair consisting of diagonal block of \(P \in GL(m,\mathbb{C})\) and a diagonal block of \(Q \in GL(n,\mathbb{C})\). The algorithm works inductively, at each step it chooses one block, transforms it to a normal form and replaces the group by the stabilizer of this normal form. The process ends in a finite number of steps, producing the canonical form. A difficulty of this approach is that the stationary subgroups of the usual normal forms with respect to the actions of \(G_{lr}[m,n]\) are not admissible and this is an obstruction for inductive steps of the algorithm. To overcome this difficulty the other normal forms, which are called modified, are introduced. This modification is useful also for other similar problems as e. g. for the reduction to a normal form of quaternion matrices by unitary similarity.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1556614
zbMATH Keywords: normal forms
zbMATH Keywords: stabilizer
zbMATH Keywords: admissible group
zbMATH Keywords: simultaneous similarity
zbMATH Keywords: matrix spaces
zbMATH Keywords: algorithm
zbMATH Keywords: canonical form
author: G. Belitskii
MaRDI profile type: Publication
cites work: Simultaneous similarity of matrices
cites work: Q3254327
cites work: Q3846750
cites work: On Unitary Equivalence
cites work: Littlewood's algorithm and quaternion matrices
cites work: Canonical forms for linear systems
cites work: Upper triangular similarity of upper triangular matrices
full work available at URL: https://doi.org/10.1007/bf01291714
OpenAlex ID: W1973531709
description: scientific article; zbMATH DE number 1556614