Solutions of the matrix linear bilateral polynomial equation and their structure
We investigate the row and column structure of solutions of the matrix polynomial equation \[ A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda), \] where \(A(\lambda)\), \(B(\lambda)\) and \(C(\lambda)\) are the matrices over the ring of polynomials \(\mathcal{F}[\lambda]\) with coefficients in f...
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| Date: | 2019 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2019
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1210 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543133030940672 |
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| author | Dzhaliuk, Nataliia S. Petrychkovych, Vasyl' M. |
| author_facet | Dzhaliuk, Nataliia S. Petrychkovych, Vasyl' M. |
| author_sort | Dzhaliuk, Nataliia S. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2019-07-14T19:54:06Z |
| description | We investigate the row and column structure of solutions of the matrix polynomial equation \[ A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda), \] where \(A(\lambda)\), \(B(\lambda)\) and \(C(\lambda)\) are the matrices over the ring of polynomials \(\mathcal{F}[\lambda]\) with coefficients in field \(\mathcal{F}\). We establish the bounds for degrees of the rows and columns which depend on degrees of the corresponding invariant factors of matrices \(A (\lambda)\) and \( B(\lambda)\). A~criterion for uniqueness of such solutions is pointed out. A method for construction of such solutions is suggested. We also established the existence of solutions of this matrix polynomial equation whose degrees are less than degrees of the Smith normal forms of matrices \(A(\lambda)\) and \( B(\lambda)\). |
| first_indexed | 2026-02-08T07:57:37Z |
| format | Article |
| id | admjournalluguniveduua-article-1210 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:57:37Z |
| publishDate | 2019 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-12102019-07-14T19:54:06Z Solutions of the matrix linear bilateral polynomial equation and their structure Dzhaliuk, Nataliia S. Petrychkovych, Vasyl' M. matrix polynomial equation, solution, polynomial matrix, semiscalar equivalence 15A21, 15A24 We investigate the row and column structure of solutions of the matrix polynomial equation \[ A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda), \] where \(A(\lambda)\), \(B(\lambda)\) and \(C(\lambda)\) are the matrices over the ring of polynomials \(\mathcal{F}[\lambda]\) with coefficients in field \(\mathcal{F}\). We establish the bounds for degrees of the rows and columns which depend on degrees of the corresponding invariant factors of matrices \(A (\lambda)\) and \( B(\lambda)\). A~criterion for uniqueness of such solutions is pointed out. A method for construction of such solutions is suggested. We also established the existence of solutions of this matrix polynomial equation whose degrees are less than degrees of the Smith normal forms of matrices \(A(\lambda)\) and \( B(\lambda)\). Lugansk National Taras Shevchenko University 2019-07-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1210 Algebra and Discrete Mathematics; Vol 27, No 2 (2019) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1210/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1210/388 Copyright (c) 2019 Algebra and Discrete Mathematics |
| spellingShingle | matrix polynomial equation solution polynomial matrix semiscalar equivalence 15A21 15A24 Dzhaliuk, Nataliia S. Petrychkovych, Vasyl' M. Solutions of the matrix linear bilateral polynomial equation and their structure |
| title | Solutions of the matrix linear bilateral polynomial equation and their structure |
| title_full | Solutions of the matrix linear bilateral polynomial equation and their structure |
| title_fullStr | Solutions of the matrix linear bilateral polynomial equation and their structure |
| title_full_unstemmed | Solutions of the matrix linear bilateral polynomial equation and their structure |
| title_short | Solutions of the matrix linear bilateral polynomial equation and their structure |
| title_sort | solutions of the matrix linear bilateral polynomial equation and their structure |
| topic | matrix polynomial equation solution polynomial matrix semiscalar equivalence 15A21 15A24 |
| topic_facet | matrix polynomial equation solution polynomial matrix semiscalar equivalence 15A21 15A24 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1210 |
| work_keys_str_mv | AT dzhaliuknataliias solutionsofthematrixlinearbilateralpolynomialequationandtheirstructure AT petrychkovychvasylm solutionsofthematrixlinearbilateralpolynomialequationandtheirstructure |