A new method based on semi-tensor product of matrices for solving reduced biquaternion matrix equation $ \sum\limits_{p = 1}^l A_pXB_p = C $ and its application in color image restoration

Abstract

In this paper, semi-tensor product of real matrices is extended to reduced biquaternion matrices, and then some new conclusions of the reduced biquaternion matrices under the vector operator are proposed using semi-tensor product of reduced biquaternion matrices, so that the reduced biquaternion matrix equation $ \sum\limits_{p = 1}^l A_pXB_p = C $ can be transformed into a reduced biquaternion linear equations, then the expression of the least squares solution of the equation is obtained using the $ \mathcal{L}_\mathcal{C} $-representation and Moore-Penrose inverse. The necessary and sufficient conditions for the compatibility and the expression of general solutions of the equation are obtained, and the minimal norm solutions are also given. Finally, our proposed method of solving the reduced biquaternion matrix equation is applied to color image restoration.

Keywords

• semi-tensor product of matrices
• reduced biquaternion matrix equation
• $ \mathcal{gh} $-representation
• $ \mathcal{l}_\mathcal{c} $-representation