Normal forms of symplectic matrices
type: Item
instance of: scholarly article
title: Normal forms of symplectic matrices
zbMATH Open document ID: 0959.15008
author: Di Dong
author: Yiming Long
publication date: 2001-04-20T00:00:00Z
review text: A basic way to study the problems related to a class of matrices is to reduce these matrices via suitable coordinate changes which preserve the basic property of the problem in consideration to a smaller family which is called the normal forms. The authors define the complete set of normal forms for real symplectic matrices possessing eigenvalues on the unit circle of the complex plane. The normal forms of symplectic matrices possessing eigenvalues away from the unit circle were considered by \textit{J. Han} and \textit{Y. Long} [Normal forms of symplectic matrices II. Research Report, Nankai Inst. of Math. Nankai Univ. (1996)]. Together with this paper the results of the present paper contain all cases for real symplectic matrices. The new interest on such normal forms comes from the study of the topological structures of the symplectic groups and their subsets, which forms the basis for defining and studying the Maslov-type index theory and its applications to linear Hamiltonian systems with periodic coefficients, and the periodic solutions of nonlinear Hamiltonian systems with Hamiltonians periodic in time [see for example \textit{Y. Long}, Pac. J. Math. 187, No. 1, 113-149 (1999; Zbl 0924.58024)].
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 37J05
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 1501750
zbMATH Keywords: normal form
zbMATH Keywords: symplectic matrix
zbMATH Keywords: eigenvalue
zbMATH Keywords: symplectic transformation
zbMATH Keywords: symplectic groups
zbMATH Keywords: Maslov-type index theory
zbMATH Keywords: linear Hamiltonian systems
zbMATH Keywords: periodic solutions
zbMATH Keywords: nonlinear Hamiltonian systems
reviewed by: Nikolai I. Osetinski
MaRDI profile type: Publication
full work available at URL: s101140050005
OpenAlex ID: W4249283750
cites work: The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems
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cites work: Canonical forms for symplectic and Hamiltonian matrices
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cites work: Q3977570
cites work: Normal forms of symplectic pencils and the discrete-time algebraic Riccati equation
DOI: 10.1007/S101140050005
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1501750