Eur. Phys. J. Plus (2013) 128: 93
https://doi.org/10.1140/epjp/i2013-13093-1

Regular Article

Generalized Householder transformations for the complex symmetric eigenvalue problem

¹ Department of Physics, Missouri University of Science and Technology, 65409, Rolla, Missouri, USA
² Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748, Garching, Germany
³ MTA-DE Particle Physics Research Group, P.O.Box 51, H-4001, Debrecen, Hungary

^* e-mail: ulj@mst.edu

Received: 1 July 2013
Accepted: 19 July 2013
Published online: 29 August 2013

Abstract

We present an intuitive and scalable algorithm for the diagonalization of complex symmetric matrices, which arise from the projection of pseudo-Hermitian and complex scaled Hamiltonians onto a suitable basis set of “trial” states. The algorithm diagonalizes complex and symmetric (non-Hermitian) matrices and is easily implemented in modern computer languages. It is based on generalized Householder transformations and relies on iterative similarity transformations T → T′ = Q ^T T Q, where Q is a complex and orthogonal, but not unitary, matrix, i.e. Q ^T = Q ⁻¹ but Q ⁺ ≠ Q ⁻¹. We present numerical reference data to support the scalability of the algorithm. We construct the generalized Householder transformations from the notion that the conserved scalar product of eigenstates Ψ _n and Ψ _m of a pseudo-Hermitian quantum mechanical Hamiltonian can be reformulated in terms of the generalized indefinite inner product ∫ dx Ψ _n (x, t) Ψ _m (x, t), where the integrand is locally defined, and complex conjugation is avoided. A few example calculations are described which illustrate the physical origin of the ideas used in the construction of the algorithm.