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 f69">$p\equiv 3(\mathrm{mod}4)$) we define the numerical range Num(M)⊆F_q²$\mathrm{Num}(M)\subseteq {\mathbb{F}}_{q^2}$ of an 0fa8bc1469f714e" title="Click to view the MathML source">n×n$n\times n$-matrix M   with coefficients in 0fbbd3c790c27ad8318997e7951" title="Click to view the MathML source">F_q²${\mathbb{F}}_{q^2}$ in terms of the usual Hermitian form. We prove that ♯(Num(M))>q$♯(\mathrm{Num}(M))>q$ (case q≠2$q\ne 2$), unless M   is unitarily equivalent to a diagonal matrix with eigenvalues contained in an affine F_q${\mathbb{F}}_q$-line. We study in details Num(M)$\mathrm{Num}(M)$ when n=2$n=2$.