Solve the linear quaternion-valued differential equations having multiple eigenvalues

The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established {see the work of Kou and Xia [Stud. Appl. Math. 141(1), 3–45 (2018)]}. They observed some profound differences between QDEs and ordinary differential equations. Also, an algorithm to evaluate the fundamental matrix by employing the eigenvalues and eigenvectors was presented in the work of Kou and Xia [Stud. Appl. Math. 141(1), 3–45 (2018)]. However, the fundamental matrix can be constructed providing that the eigenvalues are simple. If the linear system has multiple eigenvalues, how to construct the fundamental matrix? In particular, if the number of independent eigenvectors might be less than the dimension of the system, that is, the numbers of the eigenvectors are not enough to construct a fundamental matrix, how to find the “missing solutions”? The main purpose of this paper is to answer this question.