A new method based on the semi-tensor product of matrices for solving communicative quaternion matrix equation ∑i=1kAiXBi=C and its application

This paper studies the least squares problem of the commutative quaternion matrix equation (1.1), finds its minimal norm least squares (anti-)Hermitian solution. In the process of completing this work, we generalize the semi-tensor product of real matrices to the commutative quaternion matrices, then use it to extend the vector operators to the commutative quaternion matrix and propose the L-representation, which transforms the intricate commutative quaternion matrix equation into a solvable system of real linear equations, we also use GH-representation to reduce the complexity of the operation and greatly save the operation time. This can be illustrated by numerical examples in the paper. In addition, we take a special kind of commutative quaternion: reduced biquaternion as an example, and compare our method with another method in reference [33] to prove the effectiveness of our method. Finally, we apply the method used in this paper to symmetric color image restoration.