An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations
In the present paper, an iterative algorithm is proposed for solving the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi>&...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-04-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/4/776 |
| _version_ | 1797409200239804416 |
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| author | Jing Jiang Ning Li |
| author_facet | Jing Jiang Ning Li |
| author_sort | Jing Jiang |
| collection | DOAJ |
| description | In the present paper, an iterative algorithm is proposed for solving the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group of a system of quaternion matrix equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mi>A</mi><mrow><mi>l</mi><mi>s</mi></mrow></msub><msub><mi>X</mi><mi>l</mi></msub><msub><mi>B</mi><mrow><mi>l</mi><mi>s</mi></mrow></msub><mo>+</mo><msub><mi>C</mi><mrow><mi>l</mi><mi>s</mi></mrow></msub><mover accent="true"><msub><mi>X</mi><mi>l</mi></msub><mo>˜</mo></mover><msub><mi>D</mi><mrow><mi>l</mi><mi>s</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mi>F</mi><mi>s</mi></msub><mo>,</mo><mspace width="4.pt"></mspace><mi>s</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula>. A generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group, as well as the least Frobenius norm generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group, can be derived by choosing appropriate initial matrices, respectively. Moreover, the optimal approximate generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group to a given matrix group can be derived by computing the least Frobenius norm generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group of a reestablished system of matrix equations. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm. |
| first_indexed | 2024-03-09T04:10:55Z |
| format | Article |
| id | doaj.art-b2ad533bb1714ec094ad966233dd6d2b |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T04:10:55Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-b2ad533bb1714ec094ad966233dd6d2b2023-12-03T14:00:24ZengMDPI AGSymmetry2073-89942022-04-0114477610.3390/sym14040776An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix EquationsJing Jiang0Ning Li1Department of Mathematics, QiLu Normal University, Jinan 250013, ChinaSchool of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250002, ChinaIn the present paper, an iterative algorithm is proposed for solving the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group of a system of quaternion matrix equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mi>A</mi><mrow><mi>l</mi><mi>s</mi></mrow></msub><msub><mi>X</mi><mi>l</mi></msub><msub><mi>B</mi><mrow><mi>l</mi><mi>s</mi></mrow></msub><mo>+</mo><msub><mi>C</mi><mrow><mi>l</mi><mi>s</mi></mrow></msub><mover accent="true"><msub><mi>X</mi><mi>l</mi></msub><mo>˜</mo></mover><msub><mi>D</mi><mrow><mi>l</mi><mi>s</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mi>F</mi><mi>s</mi></msub><mo>,</mo><mspace width="4.pt"></mspace><mi>s</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula>. A generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group, as well as the least Frobenius norm generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group, can be derived by choosing appropriate initial matrices, respectively. Moreover, the optimal approximate generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group to a given matrix group can be derived by computing the least Frobenius norm generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-reflexive solution group of a reestablished system of matrix equations. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.https://www.mdpi.com/2073-8994/14/4/776quaternion matrix equationsiterative algorithmoptimal approximate solution |
| spellingShingle | Jing Jiang Ning Li An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations Symmetry quaternion matrix equations iterative algorithm optimal approximate solution |
| title | An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations |
| title_full | An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations |
| title_fullStr | An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations |
| title_full_unstemmed | An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations |
| title_short | An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations |
| title_sort | iterative algorithm for the generalized reflexive solution group of a system of quaternion matrix equations |
| topic | quaternion matrix equations iterative algorithm optimal approximate solution |
| url | https://www.mdpi.com/2073-8994/14/4/776 |
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