On the equivalence of polynomial matrices over a field
UDC 512.64 The polynomial $(n\times n)$ matrices $A(\lambda )$ and $B(\lambda)$ over a field ${\mathbb F}$ are  called semiscalar equivalent if there exists a nonsingular $(n\times n)$ matrix $P$ over ${\mathbb F}$ and  an invertible $(n\times n)$  polynomial m...
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| Datum: | 2024 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7926 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1865794240565477376 |
|---|---|
| author | Prokip , V. Прокіп, Володимир |
| author_facet | Prokip , V. Прокіп, Володимир |
| author_institution_txt_mv | [
{
"author": "Володимир Прокіп",
"institution": "Інститут прикладних проблем механіки і математики НАН України, Львів"
}
] |
| author_sort | Prokip , V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-11-22T08:15:25Z |
| description | UDC 512.64
The polynomial $(n\times n)$ matrices $A(\lambda )$ and $B(\lambda)$ over a field ${\mathbb F}$ are  called semiscalar equivalent if there exists a nonsingular $(n\times n)$ matrix $P$ over ${\mathbb F}$ and  an invertible $(n\times n)$  polynomial matrix $Q(\lambda )$ over ${\mathbb F[\lambda}]$ such that $A(\lambda ) = PB(\lambda )Q(\lambda )$.  We establish conditions under which nonsingular polynomial matrices $A(\lambda )$ and $B(\lambda )$  are semiscalar equivalent.  As a consequence, we present the conditions of equivalence and similarity of two sets of $(n\times n)$  matrices over an arbitrary field ${\mathbb F}.$ |
| doi_str_mv | 10.3842/umzh.v76i5.7926 |
| first_indexed | 2026-03-24T03:34:50Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7926 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:34:50Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-79262024-11-22T08:15:25Z On the equivalence of polynomial matrices over a field Про еквівалентність многочленних матриць над полем Prokip , V. Прокіп, Володимир polynomial matrix, equivalence of matrices, semi-scalar equivalence of matrices , similarity of sets of matrices Еквівалентність матриць UDC 512.64 The polynomial $(n\times n)$ matrices $A(\lambda )$ and $B(\lambda)$ over a field ${\mathbb F}$ are  called semiscalar equivalent if there exists a nonsingular $(n\times n)$ matrix $P$ over ${\mathbb F}$ and  an invertible $(n\times n)$  polynomial matrix $Q(\lambda )$ over ${\mathbb F[\lambda}]$ such that $A(\lambda ) = PB(\lambda )Q(\lambda )$.  We establish conditions under which nonsingular polynomial matrices $A(\lambda )$ and $B(\lambda )$  are semiscalar equivalent.  As a consequence, we present the conditions of equivalence and similarity of two sets of $(n\times n)$  matrices over an arbitrary field ${\mathbb F}.$ УДК 512.64 Кажуть, що многочленні $(n\times n)$-матриці $A(\lambda )$ і $B(\lambda )$ над полем ${\mathbb F}$  напівскалярно еквівалентні, якщо існують неособлива матриця $P$ над ${\mathbb F}$ та зворотна многочленна $(n\times n)$-матриця $Q(\lambda )$ над ${\mathbb F[\lambda}]$ такі, що  $A(\lambda )=PB(\lambda )Q(\lambda ).$ Встановлено умови, за яких неособливі многочленні матриці $A(\lambda )$ і $B(\lambda )$ напівскалярно еквівалентні. Як наслідок, наведено умови еквівалентності та подібності двох наборів $(n\times n)$-матриць над довільним полем ${\mathbb F}.$  Institute of Mathematics, NAS of Ukraine 2024-06-02 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7926 10.3842/umzh.v76i5.7926 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 5 (2024); 743 - 750 Український математичний журнал; Том 76 № 5 (2024); 743 - 750 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/7926/9944 Copyright (c) 2024 Володимир Михайлович Прокіп |
| spellingShingle | Prokip , V. Прокіп, Володимир On the equivalence of polynomial matrices over a field |
| title | On the equivalence of polynomial matrices over a field |
| title_alt | Про еквівалентність многочленних матриць над полем |
| title_full | On the equivalence of polynomial matrices over a field |
| title_fullStr | On the equivalence of polynomial matrices over a field |
| title_full_unstemmed | On the equivalence of polynomial matrices over a field |
| title_short | On the equivalence of polynomial matrices over a field |
| title_sort | on the equivalence of polynomial matrices over a field |
| topic_facet | polynomial matrix equivalence of matrices semi-scalar equivalence of matrices similarity of sets of matrices Еквівалентність матриць |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7926 |
| work_keys_str_mv | AT prokipv ontheequivalenceofpolynomialmatricesoverafield AT prokípvolodimir ontheequivalenceofpolynomialmatricesoverafield AT prokipv proekvívalentnístʹmnogočlennihmatricʹnadpolem AT prokípvolodimir proekvívalentnístʹmnogočlennihmatricʹnadpolem |