Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices
In this paper, we use semi-tensor product of quaternion matrices, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">L</mi></semantics></math></inline-formula...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/7/1359 |
| _version_ | 1797443444957773824 |
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| author | Xueling Fan Ying Li Zhihong Liu Jianli Zhao |
| author_facet | Xueling Fan Ying Li Zhihong Liu Jianli Zhao |
| author_sort | Xueling Fan |
| collection | DOAJ |
| description | In this paper, we use semi-tensor product of quaternion matrices, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">L</mi></semantics></math></inline-formula>-representation of quaternion matrices, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">GH</mi></semantics></math></inline-formula>-representation of special quaternion matrices such as quaternion (anti)-centrosymmetric matrices to solve the special solutions of quaternion matrix equation. Based on semi-tensor product of quaternion matrices and the structure matrix of the multiplication of quaternion, we propose the vector representation operation conclusion of quaternion matrices, and study the different matrix representations of quaternion matrices. Then the problem of the quaternion matrix equation is transformed into the corresponding problem in the real number fields by using vector representation and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">L</mi></semantics></math></inline-formula>-representation of quaternion matrices, combined with the special structure of (anti)-centrosymmetric matrices, the independent elements are extracted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">GH</mi></semantics></math></inline-formula>-representation method, so as to reduce the number of variables to be calculated and improve the calculation accuracy. Finally, the effectiveness of the method is verified by numerical examples, and the time comparison with the two existing algorithms is carried out. The algorithm in this paper is also applied in a centrosymmetric color digital image restoration model. |
| first_indexed | 2024-03-09T12:56:10Z |
| format | Article |
| id | doaj.art-a93c186f257f45edacf8113cab9493ac |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T12:56:10Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-a93c186f257f45edacf8113cab9493ac2023-11-30T21:59:46ZengMDPI AGSymmetry2073-89942022-07-01147135910.3390/sym14071359Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion MatricesXueling Fan0Ying Li1Zhihong Liu2Jianli Zhao3College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, ChinaCollege of Mathematical Sciences, Liaocheng University, Liaocheng 252000, ChinaCollege of Mathematical Sciences, Liaocheng University, Liaocheng 252000, ChinaCollege of Mathematical Sciences, Liaocheng University, Liaocheng 252000, ChinaIn this paper, we use semi-tensor product of quaternion matrices, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">L</mi></semantics></math></inline-formula>-representation of quaternion matrices, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">GH</mi></semantics></math></inline-formula>-representation of special quaternion matrices such as quaternion (anti)-centrosymmetric matrices to solve the special solutions of quaternion matrix equation. Based on semi-tensor product of quaternion matrices and the structure matrix of the multiplication of quaternion, we propose the vector representation operation conclusion of quaternion matrices, and study the different matrix representations of quaternion matrices. Then the problem of the quaternion matrix equation is transformed into the corresponding problem in the real number fields by using vector representation and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">L</mi></semantics></math></inline-formula>-representation of quaternion matrices, combined with the special structure of (anti)-centrosymmetric matrices, the independent elements are extracted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">GH</mi></semantics></math></inline-formula>-representation method, so as to reduce the number of variables to be calculated and improve the calculation accuracy. Finally, the effectiveness of the method is verified by numerical examples, and the time comparison with the two existing algorithms is carried out. The algorithm in this paper is also applied in a centrosymmetric color digital image restoration model.https://www.mdpi.com/2073-8994/14/7/1359quaternion matrix equationsemi-tensor product of quaternion matricesℒ-representation<b>GH</b>-representation(anti)-centrosymmetric matrix |
| spellingShingle | Xueling Fan Ying Li Zhihong Liu Jianli Zhao Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices Symmetry quaternion matrix equation semi-tensor product of quaternion matrices ℒ-representation <b>GH</b>-representation (anti)-centrosymmetric matrix |
| title | Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices |
| title_full | Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices |
| title_fullStr | Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices |
| title_full_unstemmed | Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices |
| title_short | Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices |
| title_sort | solving quaternion linear system based on semi tensor product of quaternion matrices |
| topic | quaternion matrix equation semi-tensor product of quaternion matrices ℒ-representation <b>GH</b>-representation (anti)-centrosymmetric matrix |
| url | https://www.mdpi.com/2073-8994/14/7/1359 |
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