# Preprint No. MPIMD/15-12

#### Author(s): Peter Benner, Heike Faßbender, Chao Yang

#### Email: benner@mpi-magdeburg.mpg.de

#### Date: 2015-07-21

#### Abstract:

The eigenproblem for complex J-symmetric matrices is considered. A proof of the existence of a transformation to the complex J-symmetric Schur form proposed in [C. Mehl. On asymptotic convergence of nonsymmetric Jacobi algorithms. SIAM J. Matrix Anal. Appl., 30:291-311, 2008.] 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 can be embedded in a *4n x 4n* real Hamiltonian matrix.

#### BibTeX:

@TECHREPORT{MPIMD15-12,

author = {Peter Benner and Heike Faßbender and Chao Yang},

title = {Some remarks on the complex J-symmetric eigenproblem},

number = {MPIMD/15-12},

month = jul,

year = 2015,

institution = {Max Planck Institute Magdeburg},

type = {Preprint},

note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}},

}