Block perturbation of symplectic matrices in Williamson's theorem

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 $S^{T}AS=D\oplus D$, 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 $\tilde{S}=SQ+\mathcal{O}(\Vert H\Vert )$, where $Q$ is a $2n\times 2n$ real symplectic as well as orthogonal matrix. Moreover, $Q$ is in $\text{symplectic block diagonal}$ 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 $\Vert \tilde{S}-S\Vert =\mathcal{O}(\Vert H\Vert )$. 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 [$\text{Linear Algebra Appl., 525:45-58, 2017}$].

13 pages, accepted in Canadian Mathematical Bulletin, the title shortened, some minor changes were made to the previous version and typos were fixed

References in corpus (5)

• Symplectic eigenvalue problem via trace minimization and Riemannian optimization
• Derivatives of symplectic eigenvalues and a Lidskii type theorem
• Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem
• Entanglement dynamics of coupled quantum oscillators in independent nonMarkovian baths
• First order sensitivity analysis of symplectic eigenvalues

Cited by in corpus (2)

• Majorization in some symplectic weak supermajorizations
• Equality in some symplectic eigenvalue inequalities