On the mixed solution of reduced biquaternion matrix equation $\sum _{i=1}^n{A}_i{X}_i{B}_i=E$ with sub-matrix constraints and its application

In this paper, we investigate the mixed solution of reduced biquaternion matrix equation $\sum _{i=1}^n{A}_i{X}_i{B}_i=E$ with sub-matrix constraints. With the help of ${\mathcal{L}}_{\mathcal{C}}$-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion matrices, the reduced biquaternion matrix equation (1.1) can be transformed into linear equations. A systematic method, $\mathcal{G}\mathcal{H}$-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the least squares problem of linear equations. Meanwhile, we give the necessary and sufficient conditions for the compatibility of reduced biquaternion matrix equation (1.1) under sub-matrix constraints. Numerical examples are given to demonstrate the results. The method proposed in this paper is applied to color image restoration.