The authors study the stability of symplectic matrices appearing in J. Williamson’s theorem [Am. J. Math. 58, 141–163 (1936; Zbl 0013.28401)], that states that for any \(2n\times 2n\) real positive definite matrix \(A\), there exists a \(2n\times 2n\) real symplectic matrix \(S\) such that \(S^\top A S = D\otimes D\), where \(D\) is an \(n\times n\) diagonal matrix with positive diagonal entries known as symplectic eigenvalues of \(A\).
Let \(A\) be a \(2n\times 2n\) real positive definite matrix and let \(H\) be an arbitrary \(2n\times 2n\) real symmetric matrix such that the perturbed matrix \(A+H\) is also positive definite. The authors show that any symplectic matrix \(\tilde S\) diagonalizing \(A+H\) in Williamson’s theorem is of the form \(\tilde S = SQ + \mathcal{O} (\|H\|)\), where \(S\) is any symplectic matrix diagonalizing \(A\) in Williamson’s theorem and \(Q\) has symplectic block diagonal form with block sizes given by twice the multiplicities of the symplectic eigenvalues of \(A\). This results in particular implies that the matrices \(S\) and \(\tilde S\) can be chosen so that \(\|\tilde S - S\| = \mathcal{O} (\|H\|)\).