On the numerical range of matrices over a finite field

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• 作者：E. Ballico¹ ; ^{ballico@science.unitn.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A33 ; 15A60
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：512
• 期：Complete
• 页码：162-171
• 全文大小：345 K

文摘

Let q be a prime power. Following a paper by Coons, Jenkins, Knowles, Luke and Rault (case q   a prime class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si1.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=838470bb17e3476be127e79e336e5f69">class="imgLazyJSB inlineImage" height="16" width="102" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516304359-si1.gif">class="mathContainer hidden">class="mathCode">$p\equiv 3(\mathrm{mod}4)$) we define the numerical range class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si2.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=9b82f034dac627f157d2f8052ff434eb" title="Click to view the MathML source">Num(M)⊆F_q²class="mathContainer hidden">class="mathCode">$\mathrm{Num}(M)\subseteq {\mathbb{F}}_{q^2}$ of an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si3.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=7412cf07dfe31aae30fa8bc1469f714e" title="Click to view the MathML source">n×nclass="mathContainer hidden">class="mathCode">$n\times n$-matrix M   with coefficients in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si21.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=07c7a0fbbd3c790c27ad8318997e7951" title="Click to view the MathML source">F_q²class="mathContainer hidden">class="mathCode">${\mathbb{F}}_{q^2}$ in terms of the usual Hermitian form. We prove that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si465.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=3aecaa4c1e40956e3d6d5ebbd3c9e51f" title="Click to view the MathML source">♯(Num(M))>qclass="mathContainer hidden">class="mathCode">$♯(\mathrm{Num}(M))>q$ (case class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si58.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=0346c3c412f54e7bd4bed578c3f1d105" title="Click to view the MathML source">q≠2class="mathContainer hidden">class="mathCode">$q\ne 2$), unless M   is unitarily equivalent to a diagonal matrix with eigenvalues contained in an affine class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si53.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=29e4987a090b74b5d87ac42c86a73d92" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$-line. We study in details class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si475.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=a06fcd918b1df86371c378cf2e6b0e98" title="Click to view the MathML source">Num(M)class="mathContainer hidden">class="mathCode">$\mathrm{Num}(M)$ when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304359&_mathId=si62.gif&_user=111111111&_pii=S0024379516304359&_rdoc=1&_issn=00243795&md5=ab066e32b501695826cb4b493f018b9b" title="Click to view the MathML source">n=2class="mathContainer hidden">class="mathCode">$n=2$.