Quaternion differential equations (QDEs) are a new kind of differential equations which differ from ordinary differential equations. Our aim is to get the exponential matrices for the QDE which is useful for finding the solution of quaternion-valued differential equations, also, we know that linear algebra is very useful to calculate the exponential for a matrix but the solution of QDE is not a linear space. Due to the noncommutativity of the quaternion, the solution set of QDE is a right free module. For this, we must read some basic concepts on Quaternions such as eigenvalues, eigenvectors, Wronskian and the difference between quaternion and complex eigenvalues and eigenvectors; by using the right eigenvalue method for quaternions we developed a fundamental matrix which is useful to construct the exponential matrices which perform a great role in solving the QDEs.

四元数微分方程（QDE）是一种不同于常微分方程的新型微分方程。我们的目标是获得 QDE 的指数矩阵，这对于找到四元值微分方程的解很有用，而且我们知道线性代数对于计算矩阵的指数非常有用，但 QDE 的解不是线性空间。由于四元数的不可交换性，QDE的解集是一个右自由模。为此，我们必须阅读有关四元数的一些基本概念，例如特征值，特征向量，Wronskian以及四元数与复特征值和特征向量之间的区别；