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.

翻译：对偶四元数矩阵在机器人学研究中具有广泛应用，其谱理论近年来受到广泛研究。本文扩展Jacobi方法以计算对偶四元数Hermitian矩阵的全部特征对，并证明其收敛性。提出采用消元策略的改进版本以减少计算时间。特别地，针对标准部分相同但对偶部分不同的特征值，我们提出一种新颖的三步Jacobi计算方法。我们证明所提三步Jacobi方法至多经过有限次迭代即可终止，并能提供特征值的$\epsilon$近似解。据我们所知，幂法与Rayleigh商迭代法均无法处理此类场景下的特征值问题。数值实验表明，所提出的Jacobi类算法具有高效性和稳定性，其性能优于幂法与Rayleigh商迭代法。