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

Jianhua Sun; Ying Li; Mingcui Zhang; Zhihong Liu; Anli Wei; Jianhua Sun; Ying Li; Mingcui Zhang; Zhihong Liu; Anli Wei

doi:10.3934/mmc.2023019

Mathematical Modelling and Control

2023, Volume 3, Issue 3: 218-232. doi: 10.3934/mmc.2023019

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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

1.
College of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, China
2.
Research Center of Semi-tensor Product of Matrices: Theory and Applications, Liaocheng, 252000, China

Received: 30 August 2022 Revised: 13 February 2023 Accepted: 13 March 2023 Published: 04 September 2023

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.
Citation: Jianhua Sun, Ying Li, Mingcui Zhang, Zhihong Liu, Anli Wei. 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[J]. Mathematical Modelling and Control, 2023, 3(3): 218-232. doi: 10.3934/mmc.2023019

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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.

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