On generalization of Williamson's theorem to real symmetric matrices

Abstract

Williamson's theorem states that if $A$ is a $2n \times 2n$ real symmetric positive definite matrix then there exists a $2n \times 2n$ real symplectic matrix $M$ such that $M^T A M=D \oplus D$, where $D$ is an $n \times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of $A$. The theorem is known to be generalized to $2n \times 2n$ real symmetric positive semidefinite matrices whose kernels are symplectic subspaces of $\mathbb{R}^{2n}$, in which case, some of the diagonal entries of $D$ are allowed to be zero. In this paper, we further generalize Williamson's theorem to $2n \times 2n$ real symmetric matrices by allowing the diagonal elements of $D$ to be any real numbers, and thus extending the notion of symplectic eigenvalues to real symmetric matrices. Also, we provide an explicit description of symplectic eigenvalues, construct symplectic matrices achieving Williamson's theorem type decomposition, and establish perturbation bounds on symplectic eigenvalues for a class of $2n \times 2n$ real symmetric matrices denoted by $\operatorname{EigSpSm}(2n)$. The set $\operatorname{EigSpSm}(2n)$ contains $2n \times 2n$ real symmetric positive semidefinite whose kernels are symplectic subspaces of $\mathbb{R}^{2n}$. Our perturbation bounds on symplectic eigenvalues for $\operatorname{EigSpSm}(2n)$ generalize known perturbation bounds on symplectic eigenvalues of positive definite matrices given by Bhatia and Jain \textit{[J. Math. Phys. 56, 112201 (2015)]}.