Minimizing numerical radius of weighted cyclic matrices under permutation of the weights

TL;DR

This paper investigates how the numerical radius of weighted cyclic matrices varies under permutations of weights, identifying the permutation that minimizes the numerical radius for matrices with fixed ordered weights.

Contribution

It provides the first analysis of the permutation that minimizes the numerical radius of weighted cyclic matrices, answering an open question posed by Chien et al.

Findings

Identifies the permutation minimizing the numerical radius for fixed weights.

Extends understanding of the numerical range behavior under permutations.

Addresses an open problem in matrix analysis regarding weighted cyclic matrices.

Abstract

In this article we answer a question asked by Chien et al. in arXiv:2304.06050 in which they study the numerical range of weighted cyclic matrices under permutation of their entries. Namely, we are interested in how $w(A_\sigma)$ fluctuates for various permutations $\sigma\in S_n$ and fixed $0\leq a_1<\cdots<a_n$ with $A_\sigma=\begin{pmatrix} 0&a_{\sigma(1)}&{}&{}&{}\cr {}&0&a_{\sigma(2)}&{}&{}\cr {}&{}&\ddots&\ddots&{}\cr {}&{}&{}&\ddots&a_{\sigma(n-1)}\cr a_{\sigma(n)}&{}&{}&{}&0 \end{pmatrix}$. Previous results of Gau \cite{gau2024proof} and Chang and Wang \cite{chang2012maximizing} made clear the case when $w(A_\sigma)$ is maximal among all the $w(A_\mu)$ with $\mu\in S_n$. Chien et al. in arXiv:2304.06050 ask what the permutation which makes $w(A_\sigma)$ minimal for $n\geq 6$ could be. Answering this question is the aim of this note.