双曲广义四元数的表示矩阵的棣莫弗定理

De Moivre’s theorem for representation matrix of hyperbolic generalized quaternions

该研究从双曲广义四元数的概念出发,首先,将双曲广义四元数的研究转化为双曲广义四元数的表示矩阵的研究;其次,利用双曲广义四元数极表示的形式,得到不同情形下双曲广义四元数的表示矩阵的棣莫弗定理,讨论了双曲广义四元数表示矩阵的方幂之间的内在联系,推广了欧拉公式;再次,给出有关双曲广义四元数的表示矩阵方程的求根公式;最后,利用算例验证了结果的正确性.

In this paper, starting from the concept of hyperbolic generalized quaternion, firstly, the study of hyperbolic generalized quaternion is transformed into the study of the matrix representation of hyperbolic generalized quaternion. Secondly, using the polar form of hyperbolic generalized quaternion, the De Moivre’s theorems for the matrix representation of hyperbolic generalized quaternion in different cases are obtained. We discuss the internal relation among the powers of the hyperbolic generalized quaternion representation matrix, and extend the Euler’s formula. Thirdly, the formula for solving the matrix representation equation of hyperbolic generalized quaternion is obtained. Finally, some examples to verify the correctness of the results are given.