Joint numerical ranges and communtativity of matrices

TL;DR

This paper explores the relationship between matrix commutativity and the geometric properties of their joint numerical ranges, providing characterizations that link algebraic and geometric aspects with implications for quantum information science.

Contribution

It establishes new criteria connecting the polyhedral nature of joint numerical ranges to the commutativity of matrices, extending understanding in matrix analysis and quantum theory.

Findings

Mutually commuting normal matrices have polyhedral joint numerical ranges for certain k

Characterization of when the c-numerical range is polyhedral for matrices

Connections between geometric properties of numerical ranges and matrix algebra

Abstract

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.