{"title":"关于k阶对称矩阵上可交换加性映射的注释","authors":"W. L. Chooi, Yean Nee Tan","doi":"10.13001/ela.2021.6349","DOIUrl":null,"url":null,"abstract":"Let $n\\geq 2$ and $1<k\\leq n$ be integers. Let $S_n(\\mathbb{F})$ be the linear space of $n\\times n$ symmetric matrices over a field $\\mathbb{F}$ of characteristic not two. In this note, we prove that an additive map $\\psi:S_n(\\mathbb{F})\\rightarrow S_n(\\mathbb{F})$ satisfies $\\psi(A)A=A\\psi(A)$ for all rank $k$ matrices $A\\in S_n(\\mathbb{F})$ if and only if there exists a scalar $\\lambda\\in \\mathbb{F}$ and an additive map $\\mu:S_n(\\mathbb{F})\\rightarrow \\mathbb{F}$ such that\\[\\psi(A)=\\lambda A+\\mu(A)I_n,\\]for all $A\\in S_n(\\mathbb{F})$, where $I_n$ is the identity matrix. Examples showing the indispensability of assumptions on the integer $k>1$ and the underlying field $\\mathbb{F}$ of characteristic not two are included.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on commuting additive maps on rank k symmetric matrices\",\"authors\":\"W. L. Chooi, Yean Nee Tan\",\"doi\":\"10.13001/ela.2021.6349\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $n\\\\geq 2$ and $1<k\\\\leq n$ be integers. Let $S_n(\\\\mathbb{F})$ be the linear space of $n\\\\times n$ symmetric matrices over a field $\\\\mathbb{F}$ of characteristic not two. In this note, we prove that an additive map $\\\\psi:S_n(\\\\mathbb{F})\\\\rightarrow S_n(\\\\mathbb{F})$ satisfies $\\\\psi(A)A=A\\\\psi(A)$ for all rank $k$ matrices $A\\\\in S_n(\\\\mathbb{F})$ if and only if there exists a scalar $\\\\lambda\\\\in \\\\mathbb{F}$ and an additive map $\\\\mu:S_n(\\\\mathbb{F})\\\\rightarrow \\\\mathbb{F}$ such that\\\\[\\\\psi(A)=\\\\lambda A+\\\\mu(A)I_n,\\\\]for all $A\\\\in S_n(\\\\mathbb{F})$, where $I_n$ is the identity matrix. Examples showing the indispensability of assumptions on the integer $k>1$ and the underlying field $\\\\mathbb{F}$ of characteristic not two are included.\",\"PeriodicalId\":50540,\"journal\":{\"name\":\"Electronic Journal of Linear Algebra\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2021-12-22\",\"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.2021.6349\",\"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.2021.6349","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
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