Block perturbation of symplectic matrices in Williamson's theorem

TL;DR

This paper investigates how small symmetric perturbations to a positive definite matrix affect its symplectic diagonalization in Williamson's theorem, providing stability results even with repeated eigenvalues.

Contribution

It extends the stability analysis of symplectic matrices in Williamson's theorem to cases with repeated eigenvalues and block perturbations.

Findings

Symplectic diagonalizers change continuously with perturbations.

The symplectic diagonalizer can be chosen close to the original.

Results hold even with repeated symplectic eigenvalues.

Abstract

Williamson's theorem states that for any $2n \times 2n$ real positive definite matrix $A$, there exists a $2n \times 2n$ real symplectic matrix $S$ such that $S^TAS=D \oplus D$, where $D$ is an $n\times n$ diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of $A$. Let $H$ be any $2n \times 2n$ real symmetric matrix such that the perturbed matrix $A+H$ is also positive definite. In this paper, we show that any symplectic matrix $\tilde{S}$ diagonalizing $A+H$ in Williamson's theorem is of the form $\tilde{S}=S Q+\mathcal{O}(\|H\|)$, where $Q$ is a $2n \times 2n$ real symplectic as well as orthogonal matrix. Moreover, $Q$ is in $\textit{symplectic block diagonal}$ form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of $A$. Consequently, we show that $\tilde{S}$ and $S$ can be chosen so that $\|\tilde{S}-S\|=\mathcal{O}(\|H\|)$. 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 [$\textit{Linear Algebra Appl., 525:45-58, 2017}$].