Block perturbation of symplectic matrices in Williamson's theorem

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 [$\textit{Linear Algebra Appl., 525:45-58, 2017}$].