Title

Diagonalization of Complex Symmetric Matrices: Generalized Householder Reflections, Iterative Deflation and Implicit Shifts

Abstract

We describe a matrix diagonalization algorithm for complex symmetric (not Hermitian) matrices, A̲=A̲T, which is based on a two-step algorithm involving generalized Householder reflections based on the indefinite inner product⟨iui &upsiloni>. This inner product is linear in both arguments and avoids complex conjugation. The complex symmetric input matrix is transformed to tridiagonal form using generalized Householder transformations (first step). An iterative, generalized QL decomposition of the tridiagonal matrix employing an implicit shift converges toward diagonal form (second step). The QL algorithm employs iterative deflation techniques when a machine-precision zero is encountered "prematurely" on the super-/sub-diagonal. The algorithm allows for a reliable and computationally efficient computation of resonance and antiresonance energies which emerge from complex-scaled Hamiltonians, and for the numerical determination of the real energy eigenvalues of pseudo-Hermitian and PT-symmetric Hamilton matrices.

Department(s)

Physics

Keywords and Phrases

Complex symmetric matrix diagonalization; Deflation techniques; Implicit shift; Indefinite inner product

International Standard Serial Number (ISSN)

0010-4655

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2017 Elsevier B.V., All rights reserved.

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