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 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 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 [Linear Algebra Appl., 525:45–58, 2017].