A note on classification of holomorphic matrices up to similarity
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 \(CNC^{-1}=N'\).
scientific article; zbMATH DE number 4202489