Singular points of the algebraic curves associated to unitary bordering matrices

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• 作者：Mao-Ting Chien^a ; ¹ ; ^{mtchien@scu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Hiroshi Nakazato^b ; ² ; ^{nakahr@hirosaki-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A60 ; 14H50
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：513
• 期：Complete
• 页码：224-239
• 全文大小：353 K

文摘

Let A   be an n×n$n\times n$ complex matrix. A ternary form associated to A   is defined as the homogeneous polynomial F_A(t,x,y)=det⁡(tI_n+xℜ(A)+yℑ(A))$F_A(t,x,y)=\det (t{I}_n+x\Re (A)+y\Im (A))$. We prove, for a unitary boarding matrix A  , the ternary form F_A(t,x,y)$F_A(t,x,y)$ is strongly hyperbolic and the algebraic curve F_A(t,x,y)=0$F_A(t,x,y)=0$ has no real singular points. As a consequence, we obtain that the higher rank numerical range of a unitary boarding matrix is strictly convex.