  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.
