  Dual quaternions and dual quaternion matrices have garnered widespread
applications in robotic research, and its spectral theory has been extensively
studied in recent years. This paper introduces the novel concept of the dual
complex adjoint matrices for dual quaternion matrices. We delve into exploring
the properties of this matrix, utilizing it to study eigenvalues of dual
quaternion matrices and defining the concept of standard right eigenvalues.
Notably, we leverage the properties of the dual complex adjoint matrix to
devise a direct solution to the Hand-Eye calibration problem. Additionally, we
apply this matrix to solve dual quaternion linear equations systems, thereby
advancing the Rayleigh quotient iteration method for computing eigenvalues of
dual quaternion Hermitian matrices, enhancing its computational efficiency.
Numerical experiments have validated the correctness of our proposed method in
solving the Hand-Eye calibration problem and demonstrated the effectiveness in
improving the Rayleigh quotient iteration method, underscoring the promising
potential of dual complex adjoint matrices in tackling dual quaternion-related
challenges.
