高阶数值域与Kippenhahn多项式
Higher-rank Numerical Ranges and Kippenhahn Polynomials
摘要：
证明了两个n × n矩阵A和B的秩k数值域$Λ_k(A)$和$Λ_k(B)$对所有k，$1\le k\le \lfloor n/2\rfloor+1$都相等，当且仅当它们的Kippenhahn多项式$p_A(x,y,z)\equiv\det(x Re A+y Im A+zI_n)$和$p_B(x,y,z)\equiv\det(x Re B+y Im B+zI_n)$重合.证明的主要工具是高秩数值域的Li-Sze特征、埃尔米特矩阵特征值的Weyl扰动定理和两个齐次多项式公共零点个数的Bezout定理。
英文摘要：
We prove that two n-by-n matrices A and B have their rank-k numerical ranges $Λ_k(A)$ and $Λ_k(B)$ equal to each other for all k, $1\le k\le \lfloor n/2\rfloor+1$, if and only if their Kippenhahn polynomials $p_A(x,y,z)\equiv\det(x Re A+y Im A+zI_n)$ and $p_B(x,y,z)\equiv\det(x Re B+y Im B+zI_n)$ coincide. The main tools for the proof are the Li-Sze characterization of higher-rank numerical ranges, Weyl's perturbation theorem for eigenvalues of Hermitian matrices and Bezout's theorem for the number of common zeros for two homogeneous polynomials.
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