Block perturbation of symplectic matrices in Williamson’s theorem
Gajendra Babu, Hemant K. Mishra · Canadian Mathematical Bulletin · 2023
Abstract Williamson’s theorem states that for any $2n \times 2n$ real positive definite matrixA, there exists a $2n \times 2n$ real symplectic matrixSsuch that $S^TAS=D \oplus D$ , whereDis an $n\times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues ofA. LetHbe 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 $\tilde {S}=S Q+\mathcal {O}(\|H\|)$ , whereQis a $2n \times 2n$ real symplectic as well as orthogonal matrix. Moreover,Qis insymplectic block diagonalform with the block sizes given by twice the multiplicities of the symplectic eigenvalues ofA. Consequently, we show that $\tilde {S}$ andScan be chosen so that $\|\tilde {S}-S\|=\mathcal {O}(\|H\|)$ . Our results hold even ifAhas 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].