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Joint numerical ranges and commutativity of matrices
Journal of Mathematical Analysis and Applications ( IF 1.3 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.jmaa.2020.124310
Chi-Kwong Li , Yiu-Tung Poon , Ya-Shu Wang

The connection between the commutativity of a family of $n\times n$ matrices and the generalized joint numerical ranges is studied. For instance, it is shown that ${\cal F}$ is a family of mutually commuting normal matrices if and only if the joint numerical range $W_k(A_1, \dots, A_m)$ is a polyhedral set for some $k$ satisfying $|n/2-k|\le 1$, where $\{A_1, \dots, A_m\}$ is a basis for the linear span of the family; equivalently, $W_k(X,Y)$ is polyhedral for any two $X, Y \in {\cal F}$. More generally, characterization is given for the $c$-numerical range $W_c(A_1, \dots, A_m)$ to be polyhedral for any $n\times n$ matrices $A_1, \dots, A_m$. Other results connecting the geometrical properties of the joint numerical ranges and the algebraic properties of the matrices are obtained. Implications of the results to representation theory, and quantum information science are discussed.

中文翻译:

矩阵的联合数值范围和交换性

研究了$n\times n$ 矩阵族的交换性与广义联合数值范围之间的联系。例如,它表明 ${\cal F}$ 是一组相互交换的正规矩阵当且仅当联合数值范围 $W_k(A_1, \dots, A_m)$ 是某个 $k$ 的多面体集满足$|n/2-k|\le 1$,其中$\{A_1, \dots, A_m\}$为族线性跨度的基;等价地,$W_k(X,Y)$ 是任意两个 $X, Y \in {\cal F}$ 的多面体。更一般地,对于任何 $n\times n$ 矩阵 $A_1​​, \dots, A_m$,$c$-数值范围 $W_c(A_1, \dots, A_m)$ 的特征是多面体。获得了连接联合数值范围的几何特性和矩阵的代数特性的其他结果。结果对表示理论的影响,
更新日期:2020-11-01
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