  Symplectic eigenvalues are conventionally defined for symmetric
positive-definite matrices via Williamson's diagonal form. Many properties of
standard eigenvalues, including the trace minimization theorem, are extended to
the case of symplectic eigenvalues. In this note, we will generalize
Williamson's diagonal form for symmetric positive-definite matrices to the case
of symmetric positive-semidefinite matrices, which allows us to define
symplectic eigenvalues, and prove the trace minimization theorem in the new
setting.
