Every 2n-by-2n complex matrix is a sum of three symplectic matrices

Let J2n=[0In−In0]. An A∈M2n(C) is called symplectic if ATJ2nA=J2n. If n=1, then we show that every matrix in M2n(C) is a sum of two symplectic matrices. If n>1, then we show that every matrix in M2n(C) is a sum of three symplectic matrices; moreover, we show that some matrices cannot be written with less than three symplectic matrices. We also show that for every A∈M2n(C), there exist symplectic P, Q∈M2n(C) and B, C, D∈Mn(C) such that PAQ=[BC0D]. If A is skew Hamiltonian (J2n−1ATJ2n=A), then we show that A is a sum of two symplectic matrices.