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Hermitian矩阵中的应用在多智能体编队控制领域具有重要意义。本文研究求解对偶四元数Hermitian矩阵右特征对的Rayleigh商迭代法。结合对偶表示，Rayleigh商迭代算法能够有效计算大规模对偶四元数Hermitian矩阵的极值特征值及其对应特征向量。进一步，推导了Rayleigh商迭代的收敛性分析，证明了其至少具有三次局部收敛速度，优于幂法线性收敛速度。数值实验表明，与求解对偶四元数Hermitian特征值问题的幂法相比，所提出的Rayleigh商迭代法具有高精度和低CPU时间成本。