Dual quaternions and dual quaternion matrices have found widespread applications in robotic research, with their spectral theory been extensively studied in recent years. This paper delves into the realm of eigenvalue computation and related problems specific to dual quaternion Hermitian matrices. We establish the connection between dual quaternion matrices and their dual complex adjoint matrices, concerning eigenvalue problems and matrix rank-k approximations. By integrating the dual complex adjoint matrix, we refine the power method for eigenvalue computation for dual quaternion Hermitian matrices, achieving greater numerical efficiency. Furthermore, leveraging the eigen-decomposition of dual complex adjoint matrices, we introduce a novel approach for calculating all eigenpairs of dual quaternion Hermitian matrices. This method surpasses the power method in terms of accuracy and speed and addresses its limitations, as exemplified by its application to the eigenvalue computation of Laplacian matrices, where our algorithm demonstrates significant advantages. Additionally, we apply the improved power method and optimal rank-k approximations to pose graph optimization problem, enhancing efficiency and success rates, especially under low observation conditions.

翻译：对偶四元数及其矩阵在机器人学研究中已得到广泛应用，其谱理论近年来亦受到广泛研究。本文深入探讨了对偶四元数埃尔米特矩阵的特征值计算及相关问题。我们建立对偶四元数矩阵与其双复伴随矩阵在特征值问题与矩阵秩-k逼近方面的联系。通过引入双复伴随矩阵，我们改进了针对对偶四元数埃尔米特矩阵特征值计算的幂法，获得了更高的数值效率。此外，基于双复伴随矩阵的特征分解，我们提出了一种计算对偶四元数埃尔米特矩阵全部特征对的新方法。该方法在精度与速度上均超越幂法，并克服了其局限性，例如在拉普拉斯矩阵特征值计算中，我们的算法展现出显著优势。另外，我们将改进的幂法与最优秩-k逼近应用于位姿图优化问题，提升了求解效率与成功率，尤其在低观测条件下效果更为突出。