Sergejchuk, V. V. A note on classification of holomorphic matrices up to similarity. (English. Russian original) Zbl 0728.15010 Funct. Anal. Appl. 25, No. 2, 135 (1991); translation from Funkts. Anal. Prilozh. 25, No. 2, 65 (1991). 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 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 \(CNC^{-1}=N'\). Reviewer: Yang Yingchen (Beijing) Cited in 1 Document MSC: 15A54 Matrices over function rings in one or more variables 15A21 Canonical forms, reductions, classification Keywords:holomorphic matrix; Arnold’s normal form; holomorphical similarity; canonical form; classification; pairs of linear operators Citations:Zbl 0259.15011 PDFBibTeX XMLCite \textit{V. V. Sergejchuk}, Funct. Anal. Appl. 25, No. 2, 135 (1991; Zbl 0728.15010); translation from Funkts. Anal. Prilozh. 25, No. 2, 65 (1991) Full Text: DOI References: [1] V. I. Arnol’d, Usp. Mat. Nauk,26, No. 2, 101-114 (1971). [2] I. M. Gel’fand and V. A. Ponomarev, Funkts. Anal. Prilozhen.,13, No. 4, 81-82 (1969). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.