  Given a real-valued positive semidefinite matrix, Williamson proved that it
can be diagonalised using symplectic matrices. The corresponding diagonal
values are known as the symplectic spectrum. This paper is concerned with the
stability of Williamson's decomposition under perturbations. We provide norm
bounds for the stability of the symplectic eigenvalues and prove that if $S$
diagonalises a given matrix $M$ to Williamson form, then $S$ is stable if the
symplectic spectrum is nondegenerate and $S^TS$ is always stable. Finally, we
sketch a few applications of the results in quantum information theory.
