Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product
This paper investigates the Sylvester-transpose matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⋉</mo><mi>X</mi><mo>+</mo><m...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/6/1094 |
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| author | Janthip Jaiprasert Pattrawut Chansangiam |
| author_facet | Janthip Jaiprasert Pattrawut Chansangiam |
| author_sort | Janthip Jaiprasert |
| collection | DOAJ |
| description | This paper investigates the Sylvester-transpose matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⋉</mo><mi>X</mi><mo>+</mo><msup><mi>X</mi><mi>T</mi></msup><mo>⋉</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⋉</mo><mi>X</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>⋉</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, and the classical Sylvester-transpose matrix equation. |
| first_indexed | 2024-03-09T22:22:49Z |
| format | Article |
| id | doaj.art-1b6afa8691a84c5497fd2a5148932218 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T22:22:49Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-1b6afa8691a84c5497fd2a51489322182023-11-23T19:10:45ZengMDPI AGSymmetry2073-89942022-05-01146109410.3390/sym14061094Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor ProductJanthip Jaiprasert0Pattrawut Chansangiam1Department of Mathematics, School of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, ThailandDepartment of Mathematics, School of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, ThailandThis paper investigates the Sylvester-transpose matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⋉</mo><mi>X</mi><mo>+</mo><msup><mi>X</mi><mi>T</mi></msup><mo>⋉</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⋉</mo><mi>X</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>⋉</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, and the classical Sylvester-transpose matrix equation.https://www.mdpi.com/2073-8994/14/6/1094Sylvester-transpose matrix equationmatrices over a fieldtensor productsemi-tensor productvector operatorlinear equations |
| spellingShingle | Janthip Jaiprasert Pattrawut Chansangiam Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product Symmetry Sylvester-transpose matrix equation matrices over a field tensor product semi-tensor product vector operator linear equations |
| title | Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product |
| title_full | Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product |
| title_fullStr | Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product |
| title_full_unstemmed | Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product |
| title_short | Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product |
| title_sort | solving the sylvester transpose matrix equation under the semi tensor product |
| topic | Sylvester-transpose matrix equation matrices over a field tensor product semi-tensor product vector operator linear equations |
| url | https://www.mdpi.com/2073-8994/14/6/1094 |
| work_keys_str_mv | AT janthipjaiprasert solvingthesylvestertransposematrixequationunderthesemitensorproduct AT pattrawutchansangiam solvingthesylvestertransposematrixequationunderthesemitensorproduct |