Block perturbation of symplectic matrices in Williamson's theorem

Article Type

Research Article

Publication Title

Canadian Mathematical Bulletin

Abstract

Williamson's theorem states that for 2n × 2n any real positive definite matrix A, there exists a 2n × 2n real symplectic matrix S such that {equation presented}, where D is an n × n diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of A. Let H be any 2n × 2n real symmetric matrix such that the perturbed matrix A + H is also positive definite. In this paper, we show that any symplectic matrix S diagonalizing A + H in Williamson's theorem is of the form {equation presented}, where Q is a 2n × 2n real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that S and S can be chosen so that {equation presented}. Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45-58, 2017].

First Page

201

Last Page

214

DOI

10.4153/S0008439523000620

Publication Date

3-15-2024

Comments

Open Access; Green Open Access; Hybrid Gold Open Access

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