An Efficient Method for Split Quaternion Matrix Equation <i>X</i> − <i>Af</i>(<i>X</i>)<i>B</i> = <i>C</i>
In this paper, we consider the split quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>−</mo><mi>A</mi><mi>f</mi><m...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/6/1158 |
| _version_ | 1797482043719811072 |
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| author | Shufang Yue Ying Li Anli Wei Jianli Zhao |
| author_facet | Shufang Yue Ying Li Anli Wei Jianli Zhao |
| author_sort | Shufang Yue |
| collection | DOAJ |
| description | In this paper, we consider the split quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>−</mo><mi>A</mi><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mi>X</mi><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mi mathvariant="normal">H</mi></msup><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">i</mi><mi mathvariant="normal">H</mi></mrow></msup><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">j</mi><mi mathvariant="normal">H</mi></mrow></msup><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">k</mi><mi mathvariant="normal">H</mi></mrow></msup><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples. |
| first_indexed | 2024-03-09T22:22:42Z |
| format | Article |
| id | doaj.art-f49e14961f314c2c864ca2ebfe6b1653 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T22:22:42Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-f49e14961f314c2c864ca2ebfe6b16532023-11-23T19:11:45ZengMDPI AGSymmetry2073-89942022-06-01146115810.3390/sym14061158An Efficient Method for Split Quaternion Matrix Equation <i>X</i> − <i>Af</i>(<i>X</i>)<i>B</i> = <i>C</i>Shufang Yue0Ying Li1Anli Wei2Jianli Zhao3Research 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 consider the split quaternion matrix equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>−</mo><mi>A</mi><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mi>X</mi><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mi mathvariant="normal">H</mi></msup><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">i</mi><mi mathvariant="normal">H</mi></mrow></msup><mo>,</mo><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">j</mi><mi mathvariant="normal">H</mi></mrow></msup><mspace width="3.33333pt"></mspace><msup><mi>X</mi><mrow><mi mathvariant="bold">k</mi><mi mathvariant="normal">H</mi></mrow></msup><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples.https://www.mdpi.com/2073-8994/14/6/1158split quaternion matrixreal representation matrixℋ representationbisymmetric matrixskew bisymmetric matrix |
| spellingShingle | Shufang Yue Ying Li Anli Wei Jianli Zhao An Efficient Method for Split Quaternion Matrix Equation <i>X</i> − <i>Af</i>(<i>X</i>)<i>B</i> = <i>C</i> Symmetry split quaternion matrix real representation matrix ℋ representation bisymmetric matrix skew bisymmetric matrix |
| title | An Efficient Method for Split Quaternion Matrix Equation <i>X</i> − <i>Af</i>(<i>X</i>)<i>B</i> = <i>C</i> |
| title_full | An Efficient Method for Split Quaternion Matrix Equation <i>X</i> − <i>Af</i>(<i>X</i>)<i>B</i> = <i>C</i> |
| title_fullStr | An Efficient Method for Split Quaternion Matrix Equation <i>X</i> − <i>Af</i>(<i>X</i>)<i>B</i> = <i>C</i> |
| title_full_unstemmed | An Efficient Method for Split Quaternion Matrix Equation <i>X</i> − <i>Af</i>(<i>X</i>)<i>B</i> = <i>C</i> |
| title_short | An Efficient Method for Split Quaternion Matrix Equation <i>X</i> − <i>Af</i>(<i>X</i>)<i>B</i> = <i>C</i> |
| title_sort | efficient method for split quaternion matrix equation i x i i af i i x i i b i i c i |
| topic | split quaternion matrix real representation matrix ℋ representation bisymmetric matrix skew bisymmetric matrix |
| url | https://www.mdpi.com/2073-8994/14/6/1158 |
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