Generalized Householder transformations for the complex symmetric eigenvalue problem

J. H. Noble, M. Lubasch, U. Jentschura

Research output: Contribution to journalArticle

6 Citations (Scopus)

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′ = QTT Q, where Q is a complex and orthogonal, but not unitary, matrix, i.e.QT = 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 Ψn and Ψm of a pseudo-Hermitian quantum mechanical Hamiltonian can be reformulated in terms of the generalized indefinite inner product ∫ dxΨn(x, t) Ψm(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.

Original languageEnglish
Article number93
JournalEuropean Physical Journal Plus
Volume128
Issue number8
DOIs
Publication statusPublished - Aug 1 2013

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Householder transformations
eigenvalues
matrices
products
conjugation
eigenvectors
projection
scalars

ASJC Scopus subject areas

  • Physics and Astronomy(all)

Cite this

Generalized Householder transformations for the complex symmetric eigenvalue problem. / Noble, J. H.; Lubasch, M.; Jentschura, U.

In: European Physical Journal Plus, Vol. 128, No. 8, 93, 01.08.2013.

Research output: Contribution to journalArticle

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