Numerical Ranges of KMS Matrices

TL;DR

This paper investigates the numerical ranges of KMS matrices, revealing conditions for their shape, boundary properties, and intersections with submatrices, enriching understanding of their spectral and geometric characteristics.

Contribution

It provides new characterizations of the numerical range of KMS matrices, including conditions for circularity, boundary line segments, and boundary intersections, which were previously unknown.

Findings

W(J_n(a)) is a circular disc only when n=2 and a≠0

Boundary contains a line segment if and only if n≥3 and |a|=1

Boundary intersections depend on matrix size, a, and principal submatrix position

Abstract

A KMS matrix is one of the form $$J_n(a)=[{array}{ccccc} 0 & a & a^2 &... & a^{n-1} & 0 & a & \ddots & \vdots & & \ddots & \ddots & a^2 & & & \ddots & a 0 & & & & 0{array}]$$ for $n\ge 1$ and $a$ in $\mathbb{C}$. Among other things, we prove the following properties of its numerical range: (1) $W(J_n(a))$ is a circular disc if and only if $n=2$ and $a\neq 0$, (2) its boundary $\partial W(J_n(a))$ contains a line segment if and only if $n\ge 3$ and $|a|=1$, and (3) the intersection of the boundaries $\partial W(J_n(a))$ and $\partial W(J_n(a)[j])$ is either the singleton $\{\min\sigma(\re J_n(a))\}$ if $n$ is odd, $j=(n+1)/2$ and $|a|>1$, or the empty set $\emptyset$ if otherwise, where, for any $n$-by-$n$ matrix $A$, $A[j]$ denotes its $j$th principal submatrix obtained by deleting its $j$th row and $j$th column ($1\le j\le n$), $\re A$ its real part $(A+A^*)/2$, and $\sigma(A)$ its spectrum.