Some remarks on the complex J-symmetric eigenproblem

The eigenproblem for complex J-symmetric matrices HC=[ACD−AT], A,C=CT, D=DT∈Cn×n is considered. A proof of the existence of a transformation to the complex J-symmetric Schur form proposed in [21] is given. The complex symplectic unitary QR decomposition and the complex symplectic SR decomposition are discussed. It is shown that a QR-like method based on the complex symplectic unitary QR decomposition is not feasible here. A complex symplectic SR algorithm is presented which can be implemented such that one step of the SR algorithm can be carried out in O(n) arithmetic operations. Based on this, a complex symplectic Lanczos method can be derived. Moreover, it is discussed how the 2n×2n complex J-symmetric matrix HC can be embedded in a 4n×4n real Hamiltonian matrix.