{"title":"CHARACTERISTIC POLYNOMIALS OF THE MATRICES WITH","authors":"HAN WANG, ZHI-WEI SUN","doi":"10.1017/s000497272400039x","DOIUrl":null,"url":null,"abstract":"<p>We determine the characteristic polynomials of the matrices <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$[q^{\\,j-k}+t]_{1\\le \\,j,k\\le n}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$[q^{\\,j+k}+t]_{1\\le \\,j,k\\le n}$</span></span></img></span></span> for any complex number <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$q\\not =0,1$</span></span></img></span></span>. As an application, for complex numbers <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$a,b,c$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$b\\not =0$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$a^2\\not =4b$</span></span></img></span></span>, and the sequence <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$(w_m)_{m\\in \\mathbb Z}$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$w_{m+1}=aw_m-bw_{m-1}$</span></span></img></span></span> for all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$m\\in \\mathbb Z$</span></span></img></span></span>, we determine the exact value of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline12.png\"><span data-mathjax-type=\"texmath\"><span>$\\det [w_{\\,j-k}+c\\delta _{jk}]_{1\\le \\,j,k\\le n}$</span></span></img></span></span>.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Australian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s000497272400039x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We determine the characteristic polynomials of the matrices $[q^{\,j-k}+t]_{1\le \,j,k\le n}$ and $[q^{\,j+k}+t]_{1\le \,j,k\le n}$ for any complex number $q\not =0,1$. As an application, for complex numbers $a,b,c$ with $b\not =0$ and $a^2\not =4b$, and the sequence $(w_m)_{m\in \mathbb Z}$ with $w_{m+1}=aw_m-bw_{m-1}$ for all $m\in \mathbb Z$, we determine the exact value of $\det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$.
期刊介绍：
Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way.
Published Bi-monthly
Published for the Australian Mathematical Society
$[q^{\,j-k}+t]_{1\le \,j,k\le n}$ and 
$[q^{\,j+k}+t]_{1\le \,j,k\le n}$ for any complex number 
$q\not =0,1$. As an application, for complex numbers 
$a,b,c$ with 
$b\not =0$ and 
$a^2\not =4b$, and the sequence 
$(w_m)_{m\in \mathbb Z}$ with 
$w_{m+1}=aw_m-bw_{m-1}$ for all 
$m\in \mathbb Z$, we determine the exact value of 
$\det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$.
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