The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with dual representation, the RQI algorithm can effectively compute the extreme eigenvalue along with the associated eigenvector of the large dual quaternion Hermitian matrices. Furthermore, a convergence analysis of the Rayleigh quotient iteration is derived, demonstrating a local convergence rate of at least cubic, which is faster than the linear convergence rate of the power method. Numerical examples are provided to illustrate the high accuracy and low CPU time cost of the proposed Rayleigh quotient iteration compared with the power method for solving the dual quaternion Hermitian eigenvalue problem.

翻译：特征值理论在对偶四元数厄米特矩阵中的应用在多智能体编队控制领域具有重要意义。本文研究了求解对偶四元数厄米特矩阵右特征对的瑞利商迭代法（RQI）。结合对偶表示，RQI算法能够高效计算大型对偶四元数厄米特矩阵的极值特征值及其对应的特征向量。此外，我们推导了瑞利商迭代法的收敛性分析，证明其具有至少三次的局部收敛速率，优于幂法的线性收敛速率。数值算例表明，与幂法求解对偶四元数厄米特特征值问题相比，所提出的瑞利商迭代法具有高精度和低CPU时间开销。