Numerical ranges over finite fields

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• 作者：Jane Ivy Coons ^{janeivycoons@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Jack Jenkins ^{jaj15@geneseo.edu" class="auth_mail" title="E-mail the corresponding author} ; Douglas Knowles ^{ddk4@geneseo.edu" class="auth_mail" title="E-mail the corresponding author} ; Rayanne A. Luke ^{ral11@geneseo.edu" class="auth_mail" title="E-mail the corresponding author} ; Patrick X. Rault ; ^{rault@geneseo.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 15A60 ; secondary ; 15A33
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：501
• 期：Complete
• 页码：37-47
• 全文大小：333 K

文摘

We explore an arithmetic analogue of the numerical range. We define the numerical range of a square matrix with entries in a finite field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300106&_mathId=si1.gif&_user=111111111&_pii=S0024379516300106&_rdoc=1&_issn=00243795&md5=1fec2c8782450979df309360e328037d" title="Click to view the MathML source">Z_p[i]class="mathContainer hidden">class="mathCode">${\mathbb{Z}}_p[i]$, for any prime p   congruent to 3 modulo 4. We establish the basic properties of these new numerical ranges, and prove several foundational results for matrices of arbitrary dimension. We classify the shapes of the numerical ranges of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300106&_mathId=si2.gif&_user=111111111&_pii=S0024379516300106&_rdoc=1&_issn=00243795&md5=bffa6a1bef2eb1e040665f139d9e6b49" title="Click to view the MathML source">2×2class="mathContainer hidden">class="mathCode">$2\times 2$ matrices over these finite fields.