The sums of symplectic, Hamiltonian, and skew-Hamiltonian matrices

A complex 2n×2n matrix A is called skew-Hamiltonian, Hamiltonian, and symplectic if AJ=A, AJ=−A, and AJ=A−1, respectively, in which J=[0In−In0] and AJ=J−1ATJ. We prove that each 2n×2n matrix is a sum of type “symplectic + Hamiltonian”. A 2n×2n matrix A is a sum of type “symplectic + symplectic” if and only if AAJ is similar to AJA. A 2n×2n matrix A is a sum of type “symplectic + skew-Hamiltonian” if and only if the Jordan blocks of A−AJ with eigenvalue 2i and size k≥ 2 come in pairs of the form Jk(2i)⊕Jk(2i) and Jk(2i)⊕Jk+1(2i).