The application of eigenvalue theory to dual quaternion Hermitian matrix holds significance in the realm of multi-agent formation control. In this paper, we focus on the numerical algorithm for the right eigenvalue of a dual quaternion Hermitian matrix. Rayleigh quotient iteration is proposed for computing the extreme eigenvalue with the associated eigenvector of the dual quaternion Hermitian matrix. We also derive an analysis of the convergence characteristics of the Rayleigh quotient iteration, which exhibits a local convergence rate of cubic. Numerical examples are provided to illustrate the efficiency of the proposed Rayleigh quotient iteration for the dual quaternion Hermitian eigenvalue problem.

翻译：对偶四元数Hermitian矩阵的特征值理论在多智能体协同控制领域中具有重要意义。本文聚焦于对偶四元数Hermitian矩阵右特征值的数值算法，提出采用Rayleigh商迭代计算该类矩阵的极值特征值及其关联特征向量。我们进一步推导了Rayleigh商迭代的收敛特性分析，证明其具有局部三次收敛速率。数值算例表明，所提出的Rayleigh商迭代方法能有效解决对偶四元数Hermitian特征值问题。