Block perturbation of symplectic matrices in Williamson’s theorem

Abstract

Williamson's theorem states that for any 2𝑛 ×2𝑛 real positive definite matrix 𝐴, there exists a 2𝑛 × 2𝑛 real symplectic matrix 𝑆 such that 𝑆 𝑇 𝐴𝑆 = 𝐷 ⊕ 𝐷, where 𝐷 is an 𝑛 × 𝑛 diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of 𝐴.Let 𝐻 be any 2𝑛 × 2𝑛 real symmetric matrix such that the perturbed matrix 𝐴 + 𝐻 is also positive definite. In this paper, we show that any symplectic matrix S diagonalizing 𝐴 + 𝐻 in Williamson's theorem is of the form S = 𝑆𝑄 + 𝒪(∥𝐻 ∥), where 𝑄 is a 2𝑛 × 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 S and 𝑆 can be chosen so that ∥ S − 𝑆∥ = 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