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.

Low-rank quaternion tensor completion method, a novel approach to recovery color videos and images is proposed in this paper. We respectively reconstruct a color image and a color video as a quaternion matrix (second-order tensor) and a third-order quaternion tensor by encoding the red, green, and blue channel pixel values on the three imaginary parts of a quaternion. Different from some traditional models which treat color pixel as a scalar and represent color channels separately, whereas, during the quaternion-based reconstruction, it is significant that the inherent color structures of color images and color videos can be completely preserved. Under the definition of Tucker rank, the global low-rank prior to quaternion tensor is encoded as the nuclear norm of unfolding quaternion matrices. Then, by applying the ADMM framework, we provide the tensor completion algorithm for any order quaternion tensors, which theoretically can be well used to recover missing entries of any multidimensional data with color structures. Simulation results for color videos and color images recovery show the superior performance and efficiency of the proposed method over some state-of-the-art existing ones.