Some remarks on the complex &ITJ&IT-symmetric eigenproblem

The eigenproblem for complex J-symmetric matrices H-C = [GRAPHICS], A, C = C-T, D = D-T is an element of C-nxn 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 x 2n complex J-symmetric matrix H-C can be embedded in a 4n x 4n real Hamiltonian matrix. (C) 2018 Elsevier Inc. All rights reserved.