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 symplectic S such $S^TAS=D \oplus D$ where D is an $n\times n$ diagonal with entries known as the eigenvalues of . Let H be symmetric perturbed $A+H$ also definite. In this paper, we show $\tilde {S}$ diagonalizing in form {S}=S Q+\mathcal {O}(\|H\|)$ Q well orthogonal matrix. Moreover, block sizes given by twice multiplicities Consequently, and can chosen so $\|\tilde {S}-S\|=\mathcal Our results hold even if has repeated eigenvalues. This generalizes stability result matrices non-repeated Idel, Gaona, Wolf [ Linear Algebra Appl., 525:45–58, 2017 ].