<p>Dual quaternion matrices play crucial roles in Hand-Eye calibration and multi-agent formation control. In this paper, we introduce the concept of dual complex adjoint matrices for dual quaternion matrices to facilitate tackling dual quaternion related problems. We construct a ring isomorphism between dual quaternion matrix ring and dual complex adjoint matrix ring. Based on such a ring isomorphism, we show that there exists a right eigenvalue of dual quaternion matrix which is a dual complex number. Moreover, such a right eigenvalue is unique under mild assumptions, which is called a representative right eigenvalue. Furthermore, leveraging this good property, we can devise a direct solution to the Hand-Eye calibration problem. Based on dual complex adjoint matrix, we also propose a novel Rayleigh quotient iteration method (DCAM-RQI) for computing eigenvalues of dual quaternion Hermitian matrices, which transforms the task of solving a dual quaternion linear equations system into solving a dual complex linear equations system, thereby improving the original Rayleigh quotient iteration method (RQI). We reveal that this advancement doubles the efficiency of RQI method. Particularly, numerical experiments on finding eigenvalues of dual quaternion Hermitian matrices arising from the multi-agent formation control show that the efficiency of DCAM-RQI is approximately quadrupled, and then effectively solve the multi-agent formation control problem.</p>