LAUSR

Abstract The eigenproblem for complex J-symmetric matrices H C = [ A C D − A T ] , A , C = C T , D = D T ∈ C n × 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 2 n × 2 n complex J-symmetric matrix H C can be embedded in a 4 n × 4 n real Hamiltonian matrix.