Generalized Jacobi Method for Computing Eigenvalues of Dual Quaternion Hermitian Matrices

TL;DR

This paper introduces a novel Jacobi method for efficiently computing eigenvalues of dual quaternion Hermitian matrices, addressing a gap where traditional methods fail, and demonstrates its effectiveness through numerical experiments.

Contribution

It extends the Jacobi method to dual quaternion Hermitian matrices, proposes an improved version with elimination, and develops a three-step algorithm with proven convergence.

Findings

The three-step Jacobi method terminates after finite iterations.

The proposed algorithms outperform power and Rayleigh quotient methods.

Numerical experiments confirm effectiveness and stability.

Abstract

Dual quaternion matrices have various applications in robotic research and its spectral theory has been extensively studied in recent years. In this paper, we extend Jacobi method to compute all eigenpairs of dual quaternion Hermitian matrices and establish its convergence. The improved version with elimination strategy is proposed to reduce the computational time. Especially, we present a novel three-step Jacobi method to compute such eigenvalues which have identical standard parts but different dual parts. We prove that the proposed three-step Jacobi method terminates after at most finite iterations and can provide $\epsilon$-approximation of eigenvalue. To the best of our knowledge, both the power method and the Rayleigh quotient iteration method can not handle such eigenvalue problem in this scenario. Numerical experiments illustrate the proposed Jacobi-type algorithms are effective and stable, and also outperform the power method and the Rayleigh quotient iteration method.