De Moivre’s Theorem for the Matrix Representation of Dual Generalized Quaternions

In this paper, based on the concept of dual generalized quaternions, the study of dual generalized quaternions is transformed into a study of the matrix representation of dual generalized quaternions. With the aid of a polar representation for dual generalized quaternions, De Moivre’s theorem is obtained for the matrix representation of dual generalized quaternions, and Euler’s formula is extended. The relations between the powers of matrices associated with dual generalized quaternions are determined, and the n-th root of the matrix representation equation of dual generalized quaternions is found.