Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

中文翻译：

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时间尺度上四元数脉冲动力学方程的柯西矩阵和刘维尔公式

摘要 在本研究中，我们通过一系列时间尺度上的四元数值函数来获得标量和矩阵指数函数。对于四元数矩阵的复伴随矩阵，建立了保证诱导矩阵为实值的充要条件。此外，给出并证明了四元数齐次和非齐次脉冲动力学方程的柯西矩阵和Liouville公式。在此基础上，还得到了它们的解的存在性、唯一性和表达式，包括它们的标量和矩阵形式。由于四元数代数是不可交换的，非四元数脉冲动力学方程的许多概念和性质是无效的，我们提供了不同时间尺度上的几个例子和反例来说明我们结果的有效性。