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 m>qm> be a prime power. Following a paper by Coons, Jenkins, Knowles, Luke and Rault (case m>q  m> a prime mmlsi1" class="mathmlsrc">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">mg 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">mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mi>pmi><mo>≡mo><mn>3mn><mspace width="0.25em">mspace><mo stretchy="false">(mo><mrow><mi mathvariant="normal">modmi>mrow><mspace width="0.25em">mspace><mn>4mn><mo stretchy="false">)mo>math>) we define the numerical range mmlsi2" class="mathmlsrc">ormulatext 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_q²mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><mrow><mi mathvariant="normal">Nummi>mrow><mo stretchy="false">(mo><mi>Mmi><mo stretchy="false">)mo><mo>⊆mo><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><msup><mrow><mi>qmi>mrow><mrow><mn>2mn>mrow>msup>mrow>msub>math> of an mmlsi3" class="mathmlsrc">ormulatext 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&times;nmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><mi>nmi><mo>&times;mo><mi>nmi>math>-matrix m>M  m> with coefficients in mmlsi21" class="mathmlsrc">ormulatext 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_q²mathContainer hidden">mathCode"><math altimg="si21.gif" overflow="scroll"><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><msup><mrow><mi>qmi>mrow><mrow><mn>2mn>mrow>msup>mrow>msub>math> in terms of the usual Hermitian form. We prove that mmlsi465" class="mathmlsrc">ormulatext 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))>qmathContainer hidden">mathCode"><math altimg="si465.gif" overflow="scroll"><mo>♯mo><mo stretchy="false">(mo><mrow><mi mathvariant="normal">Nummi>mrow><mo stretchy="false">(mo><mi>Mmi><mo stretchy="false">)mo><mo stretchy="false">)mo><mo>>mo><mi>qmi>math> (case mmlsi58" class="mathmlsrc">ormulatext 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≠2mathContainer hidden">mathCode"><math altimg="si58.gif" overflow="scroll"><mi>qmi><mo>≠mo><mn>2mn>math>), unless m>M  m> is unitarily equivalent to a diagonal matrix with eigenvalues contained in an affine mmlsi53" class="mathmlsrc">ormulatext 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_qmathContainer hidden">mathCode"><math altimg="si53.gif" overflow="scroll"><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><mi>qmi>mrow>msub>math>-line. We study in details mmlsi475" class="mathmlsrc">ormulatext 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)mathContainer hidden">mathCode"><math altimg="si475.gif" overflow="scroll"><mrow><mi mathvariant="normal">Nummi>mrow><mo stretchy="false">(mo><mi>Mmi><mo stretchy="false">)mo>math> when mmlsi62" class="mathmlsrc">ormulatext 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=2mathContainer hidden">mathCode"><math altimg="si62.gif" overflow="scroll"><mi>nmi><mo>=mo><mn>2mn>math>.