Several observations on symplectic, Hamiltonian, and skew-Hamiltonian matrices
type: Item
instance of: scholarly article
title: Several observations on symplectic, Hamiltonian, and skew-Hamiltonian matrices
zbMATH Open document ID: 1077.15010
published in: Linear Algebra and its Applications
publication date: 2005-06-01T00:00:00Z
review text: There is a classical theorem relating strict equivalence and \(T\)-congruence between pencils of complex symmetric or skew-symmetric matrices. In this paper the authors prove the following version of this theorem for Hamiltonian or skew-Hamiltonian matrices: Let \(A,B,C,D\) be matrices in the set \(M_{2m}(\mathbb{C})\) of all complex \(2m\times 2m\) matrices such that one of the following conditions is satisfied: (i) all four matrices are Hamiltonian; or (ii) all four matrices are skew-symmetric Hamiltonian; or (iii) \(A\) and \(C\) are Hamiltonian, while \(B\) and \(D\) are skew-symmetric Hamiltonian, or vice versa. Then, if the pencils \(A+ \lambda B\) and \(C+\lambda D\) are strictly equivalent, they are \(J\)-congruent. Moreover, a symplectic counterpart of a result of \textit{H. Xu} [Linear Algebra Appl. 368, 1--24 (2003; Zbl 1025.15016)] concerning the singular value decomposition of a conjugate symplectic matrix is shown: Every symplectic matrix \(S\in M_{2m}(\mathbb{C})\) admits a singular value decomposition \(S=U\Sigma V^*\), where all of the three factors \(U, \Sigma\) and \(V\) are symplectic. Finally the authors discuss some implications which can be derived from \textit{K. Veselić}'s result on definite pairs of Hermitian matrices [Numer. Math. 64, No. 2, 241--269 (1993; Zbl 0805.65038)] in the case of skew-Hermitian matrices: Let \(A\) be a \(J\)-skew-Hermitian matrix such that the Hermitian pair \((iI,iJ)\) is definite. All conjugate symplectic matrices \(Y\) which bring \(A\) through similarity transformations to the special form \(\left(\begin{smallmatrix} D_1 & iD_2\\ -iD_2 & D_1\end{smallmatrix}\right)\) have the same spectral (or Frobenius) condition number.
reviewed by: Huberta Lausch
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A12
zbMATH DE Number: 2173014
zbMATH Keywords: Hamiltonian matrix
zbMATH Keywords: skew Hamiltonian matrix
zbMATH Keywords: conjugate symplectic matrix
zbMATH Keywords: \(J\)-Hermitian matrix
zbMATH Keywords: \(J\)-skew-Hermitian matrix
zbMATH Keywords: singular value decomposition
zbMATH Keywords: definite Hermitian pair
zbMATH Keywords: pencils of matrices
zbMATH Keywords: symplectic matrix
zbMATH Keywords: similarity transformations
zbMATH Keywords: condition number
zbMATH Keywords: strict equivalence
zbMATH Keywords: \(T\)-congruence
zbMATH Keywords: factorization
author: Heike Faßbender
author: Kh. D. Ikramov
MaRDI profile type: Publication
full work available at URL: j.laa.2004.12.012
OpenAlex ID: W2073979097
cites work: Q3254327
cites work: Computing the Polar Decomposition and the Matrix Sign Decomposition in Matrix Groups
cites work: Matrix Analysis
cites work: Structured tools for structured matrices
cites work: A Jacobi eigenreduction algorithm for definite matrix pairs
cites work: An SVD-like matrix decomposition and its applications
DOI: 10.1016/J.LAA.2004.12.012
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description: scientific article; zbMATH DE number 2173014