On the numerical range of matrices over a finite field

详细信息    查看全文

• 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 d="mmlsi1" class="mathmlsrc">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">dth="102" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516304359-si1.gif">dden">de">mi>pmi>≡3dth="0.25em">(<mi mathvariant="normal">modmi>dth="0.25em">4)) we define the numerical range d="mmlsi2" class="mathmlsrc">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_q²dden">de">mi mathvariant="normal">Nummi>(<mi>Mmi>)⊆<mi mathvariant="double-struck">Fmi><mi>qmi>2 of an d="mmlsi3" class="mathmlsrc">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×ndden">de">mi>nmi>×<mi>nmi>-matrix M   with coefficients in d="mmlsi21" class="mathmlsrc">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_q²dden">de">mi mathvariant="double-struck">Fmi><mi>qmi>2 in terms of the usual Hermitian form. We prove that d="mmlsi465" class="mathmlsrc">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))>qdden">de">mi mathvariant="normal">Nummi>(<mi>Mmi>))><mi>qmi> (case d="mmlsi58" class="mathmlsrc">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≠2dden">de">mi>qmi>≠2), unless M   is unitarily equivalent to a diagonal matrix with eigenvalues contained in an affine d="mmlsi53" class="mathmlsrc">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_qdden">de">mi mathvariant="double-struck">Fmi><mi>qmi>-line. We study in details d="mmlsi475" class="mathmlsrc">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)dden">de">mi mathvariant="normal">Nummi>(<mi>Mmi>) when d="mmlsi62" class="mathmlsrc">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=2dden">de">mi>nmi>=2.