Applications of Dual Complex Adjoint Matrix in Eigenvalue Computation of Dual Quaternion Hermitian Matrix

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.