Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions
We consider when the quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mi>B</mi><mo>+</mo><mi>C</mi>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/1/40 |
| _version_ | 1797436928245628928 |
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| author | Xin Liu Kaiqi Wen Yang Zhang |
| author_facet | Xin Liu Kaiqi Wen Yang Zhang |
| author_sort | Xin Liu |
| collection | DOAJ |
| description | We consider when the quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mi>B</mi><mo>+</mo><mi>C</mi><mi>X</mi><mi>D</mi><mo>=</mo><mi>E</mi></mrow></semantics></math></inline-formula> has a reflexive (or anti-reflexive) solution with respect to a given generalized reflection matrix. We adopt a real representation method to derive the solutions when it is solvable. Moreover, we obtain the explicit expressions of the least-squares reflexive (or anti-reflexive) solutions. |
| first_indexed | 2024-03-09T11:09:38Z |
| format | Article |
| id | doaj.art-fae23b69c4064846a9af09a66b33648f |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T11:09:38Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-fae23b69c4064846a9af09a66b33648f2023-12-01T00:50:51ZengMDPI AGSymmetry2073-89942022-12-011514010.3390/sym15010040Matrix Equation’s Reflexive and Anti-Reflexive Solutions over QuaternionsXin Liu0Kaiqi Wen1Yang Zhang2Macau Institute of Systems Engineering, Faculty of Innovation Engineering, Macau University of Science and Technology, Avenida Wai Long, Macau 999078, ChinaMacau Institute of Systems Engineering, Faculty of Innovation Engineering, Macau University of Science and Technology, Avenida Wai Long, Macau 999078, ChinaDepartment of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, CanadaWe consider when the quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>X</mi><mi>B</mi><mo>+</mo><mi>C</mi><mi>X</mi><mi>D</mi><mo>=</mo><mi>E</mi></mrow></semantics></math></inline-formula> has a reflexive (or anti-reflexive) solution with respect to a given generalized reflection matrix. We adopt a real representation method to derive the solutions when it is solvable. Moreover, we obtain the explicit expressions of the least-squares reflexive (or anti-reflexive) solutions.https://www.mdpi.com/2073-8994/15/1/40reflexive matrixanti-reflexive matrixquaternion matrix equationreal representation |
| spellingShingle | Xin Liu Kaiqi Wen Yang Zhang Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions Symmetry reflexive matrix anti-reflexive matrix quaternion matrix equation real representation |
| title | Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions |
| title_full | Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions |
| title_fullStr | Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions |
| title_full_unstemmed | Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions |
| title_short | Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions |
| title_sort | matrix equation s reflexive and anti reflexive solutions over quaternions |
| topic | reflexive matrix anti-reflexive matrix quaternion matrix equation real representation |
| url | https://www.mdpi.com/2073-8994/15/1/40 |
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