Several observations on symplectic, Hamiltonian, and skew-Hamiltonian matrices

We prove a Hamiltonian/skew-Hamiltonian version of the classical theorem relating strict equivalence and T-congruence between pencils of complex symmetric or skew-symmetric matrices. Then, we give a pure symplectic variant of the recent result of Xu concerning the singular value decomposition of a conjugate symplectic matrix. Finally, we discuss implications that can be derived from Veselić's result on definite pairs of Hermitian matrices for the skew-Hamiltonian situation.