A note on classification of holomorphic matrices up to similarity
type: Item
instance of: scholarly article
title: A note on classification of holomorphic matrices up to similarity
zbMATH Open document ID: 0728.15010
DOI: 10.1007/BF01079595
author: V. V. Sergejchuk
publication date: 1991-01-01T00:00:00Z
review text: In this remark the author points out that the problem of canonical form of the first two matrices \(A_ 0,A_ 1\) in the expansion \(A(x)=A_ 0+xA_ 1+...\) is ``wild'', i.e., it contains the classical unsolved problem of classification of pairs of linear operators in finite dimensional vector spaces, and consequently the problem of classification of any system of vectors. It is proved that the two matrices in \textit{V. I. Arnol'd}'s normal form [Usp. Mat. Nauk 26, No.2 (158), 101-114 (1971; Zbl 0259.15011)] \[ A(x) = \begin{pmatrix} 0 & E & 0 & 0 \\ 0 & 0 & E & 0 \\ 0 & \chi M & 0 & \chi N \\ \chi E & 0 & 0 & 0 \end{pmatrix}, \quad B(x) = \begin{pmatrix} 0 & E & 0 & 0 \\ 0 & 0 & E & 0 \\ 0 & \chi M'& 0 & \chi N' \\ \chi E & 0 & 0 & 0 \end{pmatrix} \] (where the square blocks \(M,N,M',N'\) are all constant matrices) are holomorphically similar if and only if there exists a constant matrix such that \(CMC^{-1}=M'\) and \(CNC^{-1}=N'\).
reviewed by: Yang Yingchen
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 4202489
zbMATH Keywords: holomorphic matrix
zbMATH Keywords: Arnold's normal form
zbMATH Keywords: holomorphical similarity
zbMATH Keywords: canonical form
zbMATH Keywords: classification
zbMATH Keywords: pairs of linear operators
MaRDI profile type: Publication
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4202489