论文标题
矩阵的关节数值范围和沟通
Joint numerical ranges and communtativity of matrices
论文作者
论文摘要
研究了$ n \ times n $矩阵的通勤性与广义关节数值范围之间的联系。 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家庭的跨度；同等地，对于任何两个$ x，y \ in {\ cal f} $，$ w_k（x，y）$都是多面的。更一般而言，为$ c $ -numerical范围$ w_c（a_1，\ dots，a_m）$给出了表征，用于任何$ n \ times n $ n $矩阵$ a_1，\ dots，a_m $。获得了连接关节数值范围的几何特性和矩阵的代数特性的结果。讨论了结果对表示理论和量子信息科学的含义。
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.
