On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application
In this paper, we investigate the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints. With the help of $ \mathcal{L_C} $-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion ma...
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AIMS Press
2023-10-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231427?viewType=HTML |
| _version_ | 1797644887833706496 |
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| author | Yimeng Xi Zhihong Liu Ying Li Ruyu Tao Tao Wang |
| author_facet | Yimeng Xi Zhihong Liu Ying Li Ruyu Tao Tao Wang |
| author_sort | Yimeng Xi |
| collection | DOAJ |
| description | In this paper, we investigate the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints. With the help of $ \mathcal{L_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{GH} $-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. |
| first_indexed | 2024-03-11T14:37:55Z |
| format | Article |
| id | doaj.art-4d1c8f88769f41f782d7d77778707500 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-11T14:37:55Z |
| publishDate | 2023-10-01 |
| publisher | AIMS Press |
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| series | AIMS Mathematics |
| spelling | doaj.art-4d1c8f88769f41f782d7d777787075002023-10-31T01:20:58ZengAIMS PressAIMS Mathematics2473-69882023-10-01811279012792310.3934/math.20231427On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its applicationYimeng Xi0Zhihong Liu1Ying Li2Ruyu Tao3Tao Wang4Research Center of Semi-tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, ChinaResearch Center of Semi-tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, ChinaResearch Center of Semi-tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, ChinaResearch Center of Semi-tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, ChinaResearch Center of Semi-tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, ChinaIn this paper, we investigate the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints. With the help of $ \mathcal{L_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{GH} $-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the least squares problem of linear equations. Meanwhile, we give the necessary and sufficient conditions for the compatibility of reduced biquaternion matrix equation (1.1) under sub-matrix constraints. Numerical examples are given to demonstrate the results. The method proposed in this paper is applied to color image restoration.https://www.aimspress.com/article/doi/10.3934/math.20231427?viewType=HTMLreduced biquaternionsub-matrix constraints$ \mathcal{gh} $-representation$ \mathcal{l_c} $-representation |
| spellingShingle | Yimeng Xi Zhihong Liu Ying Li Ruyu Tao Tao Wang On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application AIMS Mathematics reduced biquaternion sub-matrix constraints $ \mathcal{gh} $-representation $ \mathcal{l_c} $-representation |
| title | On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application |
| title_full | On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application |
| title_fullStr | On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application |
| title_full_unstemmed | On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application |
| title_short | On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application |
| title_sort | on the mixed solution of reduced biquaternion matrix equation sum limits i 1 na ix ib i e with sub matrix constraints and its application |
| topic | reduced biquaternion sub-matrix constraints $ \mathcal{gh} $-representation $ \mathcal{l_c} $-representation |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231427?viewType=HTML |
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