Complementary bases in symplectic matrices and a proof that their determinant is one
type: Item
instance of: scholarly article
title: Complementary bases in symplectic matrices and a proof that their determinant is one
zbMATH Open document ID: 1112.15002
author: Froilán M. Dopico
author: Charles R. Johnson
published in: Linear Algebra and its Applications
publication date: 2007-02-14T00:00:00Z
review text: It is well known that symplectic transformations have determinant \(+1\). E.g., this follows immediately from the fact that the symplectic group is generated by symplectic transvections, but various other proofs are known. The authors give a very short direct proof for the fact the determinant of a symplectic matrix is \(+1\) under the extra assumption that the given matrix has a certain block form with one invertible block. Next, linearly independent rows and columns of a symplectic matrix are exhibited which then allows to reduce the case of an arbitrary symplectic matrix to the particular case mentioned in the above.
reviewed by: Hans Havlicek
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 5126044
zbMATH Keywords: symplectic matrix
zbMATH Keywords: complementary bases
zbMATH Keywords: patterns of zeros
zbMATH Keywords: Schur complement
zbMATH Keywords: determinant
zbMATH Keywords: symplectic transformations
MaRDI profile type: Publication
full work available at URL: j.laa.2006.06.014
OpenAlex ID: W1975857703
cites work: Q4174811
cites work: Q4167627
cites work: A Chart of Numerical Methods for Structured Eigenvalue Problems
cites work: Symplectic methods for the symplectic eigenproblem
cites work: Matrix Analysis
cites work: Canonical forms for Hamiltonian and symplectic matrices and pencils
cites work: Structured tools for structured matrices
cites work: Corrigenda to: Introduction to Symplectic Topology
cites work: The autonomous linear quadratic control problem. Theory and numerical solution
DOI: 10.1016/J.LAA.2006.06.014
description: scientific article; zbMATH DE number 5126044