{"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}
引用次数: 0
Abstract
Let $n\geq 2$ and $11$ and the underlying field $\mathbb{F}$ of characteristic not two are included.
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