The eigenstructure and Jordan form of the Fourier transform over fields of characteristic 2 and a generalized Vandermonde-type formula

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• 作者：M. Barone^a ; ^{marco@dmat.ufpe.br" class="auth_mail" title="E-mail the corresponding author} ; J.B. Lima^b ; ^{juliano_bandeira@ieee.org" class="auth_mail" title="E-mail the corresponding author} ; R.M. Campello de Souza^b ; ^{ricardo@ufpe.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：12E20 ; 15A18 ; 15B33 ; 43A32
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：494
• 期：Complete
• 页码：245-262
• 全文大小：374 K

文摘

In this paper, we describe the eigenstructure and the Jordan form of the Fourier transform matrix generated by a primitive N  -th root of unity in a field of characteristic 2. We find that the only eigenvalue is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515007582&_mathId=si1.gif&_user=111111111&_pii=S0024379515007582&_rdoc=1&_issn=00243795&md5=73abf8c793461eb250f8d4f994637d28" title="Click to view the MathML source">λ=1class="mathContainer hidden">class="mathCode">$\lambda =1$ and its eigenspace has dimension class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515007582&_mathId=si2.gif&_user=111111111&_pii=S0024379515007582&_rdoc=1&_issn=00243795&md5=7185dfc8e4be151f19cfb8a9d0b93e42">class="imgLazyJSB inlineImage" height="20" width="51" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379515007582-si2.gif">class="mathContainer hidden">class="mathCode">$[\frac{N}{4}]+1$; we provide a basis of eigenvectors and a Jordan basis. The problem has already been solved, for number theoretic transforms, in any other finite characteristic. However, in characteristic 2 classical results about geometric multiplicity do not apply and we have to resort to different techniques in order to determine a basis of eigenvectors and a Jordan basis. We make use of a modified version of the Vandermonde's formula, which applies to matrices whose entries are powers of elements of the form class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515007582&_mathId=si3.gif&_user=111111111&_pii=S0024379515007582&_rdoc=1&_issn=00243795&md5=aa10e5dec4f7628a23761b1fe945344c" title="Click to view the MathML source">x+x⁻¹class="mathContainer hidden">class="mathCode">$x+x^{-1}$.