  We obtain several analogs of real polar decomposition for even dimensional
matrices. In particular, we decompose a non-degenerate matrix as a product of a
Hamiltonian and an anti-symplectic matrix and under additional requirements we
decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our
results to study bosonic Gaussian channels up to inhomogeneous symplectic
transforms.
