Matrix diagonalisation in sesquilinear symplectic spaces

The symplectic inner product on $C^{2n}$ is the sesquilinear form given by$[x,y]=〈x,J_{2n}y〉,$ where $J_{2n}$ is the real skew-symmetric, orthogonal $2\times 2$ block matrix $\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right]$. We derive results analogous to the spectral theorem and singular value decomposition for complex matrices such as Hamiltonian and J-normal matrices, in the sesquilinear symplectic inner product spaces.