On Simultaneous Symplectic Diagonalization in the sense of Williamson's Theorem

TL;DR

This paper extends Williamson's theorem to specific classes of symmetric matrices, providing conditions for their simultaneous symplectic diagonalization and applying these results to physical systems and phase space geometry.

Contribution

It generalizes Williamson's theorem to symmetric matrices with negative index 1 and positive semi-definite matrices, establishing new conditions for simultaneous symplectic diagonalization.

Findings

Conditions for simultaneous symplectic diagonalization of certain symmetric matrices

Application to phase space constraints and symplectic capacities

Connections between degenerate and non-degenerate decompositions

Abstract

Williamson's theorem is well known for symmetric matrices. In this paper, we state and re-derive some of the cases of Williamson's theorem for symmetric positive-semi definite matrices and symmetric matrices having negative index 1, due to H\"ormander. We prove theorems that guarantee conditions under which two symmetric positive-definite matrices can be simultaneously diagonalized in the sense of Williamson's theorem and their corollaries. Finally, we provide an application of this result to physical systems and another connecting the decompositions for the degenerate and non-degenerate cases, involving phase space constraints that we later apply to phase space cylinders and ellipsoids via symplectic capacities.