Generalized Householder Transformations for the Complex Symmetric Eigenvalue Problem

TL;DR

This paper introduces a scalable algorithm based on generalized Householder transformations for diagonalizing complex symmetric matrices, relevant in pseudo-Hermitian quantum mechanics, avoiding complex conjugation and demonstrating numerical effectiveness.

Contribution

The paper presents a novel, scalable algorithm for complex symmetric matrix diagonalization using generalized Householder transformations, applicable to pseudo-Hermitian quantum systems.

Findings

Supports scalability with numerical data

Constructs transformations from indefinite inner products

Illustrates with example calculations

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 equals Q^(-1) but Q^+ is different from 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 Psi_n and Psi_m of a pseudo-Hermitian quantum mechanical Hamiltonian can be reformulated in terms of the generalized indefinite inner product [integral of the product Psi_n(x,t) Psi_m(x,t) over dx], 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.