Spectral properties of small Hadamard matrices

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• 作者：Dorin Ervin Dutkay^a ; ^{Dorin.Dutkay@ucf.edu" class="auth_mail" title="E-mail the corresponding author} ; John Haussermann^a ; ^{jhaussermann@knights.ucf.edu" class="auth_mail" title="E-mail the corresponding author} ; Eric Weber^b ; ^{esweber@iastate.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05B20 ; 65T50 ; 11L05
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：506
• 期：Complete
• 页码：363-381
• 全文大小：369 K

文摘

We prove that if A and B   are Hadamard matrices which are both of size class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516302233&_mathId=si1.gif&_user=111111111&_pii=S0024379516302233&_rdoc=1&_issn=00243795&md5=19618668479376adc0a75fd1f8a8748d" title="Click to view the MathML source">4×4class="mathContainer hidden">class="mathCode">$4\times 4$ or class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516302233&_mathId=si2.gif&_user=111111111&_pii=S0024379516302233&_rdoc=1&_issn=00243795&md5=48c8d01a6b292778a1af5b0a690bac0a" title="Click to view the MathML source">5×5class="mathContainer hidden">class="mathCode">$5\times 5$ and in dephased form, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516302233&_mathId=si153.gif&_user=111111111&_pii=S0024379516302233&_rdoc=1&_issn=00243795&md5=a0d3fdb5fd2fe43bb0f45ad1aa7e50a2" title="Click to view the MathML source">tr(A)=tr(B)class="mathContainer hidden">class="mathCode">$tr(A)=tr(B)$ implies that A and B have the same eigenvalues, including multiplicity. We calculate explicitly the spectrum for these matrices. We also extend these results to larger Hadamard matrices which are permutations of the Fourier matrix and calculate their spectral multiplicities.