Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem

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, have been extended to case symplectic eigenvalues. In this note, we will generalize form positive-semidefinite matrices, which allows us define and prove theorem in new setting.