{"title":"关于多项式矩阵半标量等价的一个注记","authors":"V. Prokip","doi":"10.13001/ela.2022.6505","DOIUrl":null,"url":null,"abstract":"Polynomial matrices $A(\\lambda)$ and $B(\\lambda)$ of size $n\\times n$ over a field $\\mathbb {F}$ are semiscalar equivalent if there exist a nonsingular $n\\times n$ matrix $P$ over $\\mathbb F$ and an invertible $n\\times n$ matrix $Q(\\lambda)$ over $\\mathbb F[\\lambda]$ such that $A(\\lambda)=PB(\\lambda)Q(\\lambda)$. The aim of this article is to present necessary and sufficient conditions for the semiscalar equivalence of nonsingular matrices $A(\\lambda)$ and $ B(\\lambda) $ over a field ${\\mathbb F }$ of characteristic zero in terms of solutions of a homogenous system of linear equations.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on semiscalar equivalence of polynomial matrices\",\"authors\":\"V. Prokip\",\"doi\":\"10.13001/ela.2022.6505\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Polynomial matrices $A(\\\\lambda)$ and $B(\\\\lambda)$ of size $n\\\\times n$ over a field $\\\\mathbb {F}$ are semiscalar equivalent if there exist a nonsingular $n\\\\times n$ matrix $P$ over $\\\\mathbb F$ and an invertible $n\\\\times n$ matrix $Q(\\\\lambda)$ over $\\\\mathbb F[\\\\lambda]$ such that $A(\\\\lambda)=PB(\\\\lambda)Q(\\\\lambda)$. The aim of this article is to present necessary and sufficient conditions for the semiscalar equivalence of nonsingular matrices $A(\\\\lambda)$ and $ B(\\\\lambda) $ over a field ${\\\\mathbb F }$ of characteristic zero in terms of solutions of a homogenous system of linear equations.\",\"PeriodicalId\":50540,\"journal\":{\"name\":\"Electronic Journal of Linear Algebra\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2022-03-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Journal of Linear Algebra\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.13001/ela.2022.6505\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Linear Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.13001/ela.2022.6505","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
A note on semiscalar equivalence of polynomial matrices
Polynomial matrices $A(\lambda)$ and $B(\lambda)$ of size $n\times n$ over a field $\mathbb {F}$ are semiscalar equivalent if there exist a nonsingular $n\times n$ matrix $P$ over $\mathbb F$ and an invertible $n\times n$ matrix $Q(\lambda)$ over $\mathbb F[\lambda]$ such that $A(\lambda)=PB(\lambda)Q(\lambda)$. The aim of this article is to present necessary and sufficient conditions for the semiscalar equivalence of nonsingular matrices $A(\lambda)$ and $ B(\lambda) $ over a field ${\mathbb F }$ of characteristic zero in terms of solutions of a homogenous system of linear equations.
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