Generalized Householder transformations for the complex symmetric eigenvalue problem

  • J. H. Noble
  • M. Lubasch
  • U. D. Jentschura
Regular Article


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 TT′ = 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.


Machine Accuracy Anharmonic Oscillator Tridiagonal Matrix Generalize Householder Tridiagonal Form 
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Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of PhysicsMissouri University of Science and TechnologyRollaUSA
  2. 2.Max-Planck-Institut für QuantenoptikGarchingGermany
  3. 3.MTA-DE Particle Physics Research GroupDebrecenHungary

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