Minimizing numerical radius of weighted cyclic matrices under permutation of the weights
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866915697838260224 |
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| author | Marionnet, Simon |
| author_facet | Marionnet, Simon |
| contents | 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 $w(A_σ)$ fluctuates for various permutations $σ\in S_n$ and fixed $0\leq a_1<\cdots<a_n$ with $A_σ=\begin{pmatrix} 0&a_{σ(1)}&{}&{}&{}\cr {}&0&a_{σ(2)}&{}&{}\cr {}&{}&\ddots&\ddots&{}\cr {}&{}&{}&\ddots&a_{σ(n-1)}\cr a_{σ(n)}&{}&{}&{}&0 \end{pmatrix}$. Previous results of Gau \cite{gau2024proof} and Chang and Wang \cite{chang2012maximizing} made clear the case when $w(A_σ)$ is maximal among all the $w(A_μ)$ with $μ\in S_n$. Chien et al. in arXiv:2304.06050 ask what the permutation which makes $w(A_σ)$ minimal for $n\geq 6$ could be. Answering this question is the aim of this note. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_17399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimizing numerical radius of weighted cyclic matrices under permutation of the weights Marionnet, Simon Functional Analysis 15A60, 15B99 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 $w(A_σ)$ fluctuates for various permutations $σ\in S_n$ and fixed $0\leq a_1<\cdots<a_n$ with $A_σ=\begin{pmatrix} 0&a_{σ(1)}&{}&{}&{}\cr {}&0&a_{σ(2)}&{}&{}\cr {}&{}&\ddots&\ddots&{}\cr {}&{}&{}&\ddots&a_{σ(n-1)}\cr a_{σ(n)}&{}&{}&{}&0 \end{pmatrix}$. Previous results of Gau \cite{gau2024proof} and Chang and Wang \cite{chang2012maximizing} made clear the case when $w(A_σ)$ is maximal among all the $w(A_μ)$ with $μ\in S_n$. Chien et al. in arXiv:2304.06050 ask what the permutation which makes $w(A_σ)$ minimal for $n\geq 6$ could be. Answering this question is the aim of this note. |
| title | Minimizing numerical radius of weighted cyclic matrices under permutation of the weights |
| topic | Functional Analysis 15A60, 15B99 |
| url | https://arxiv.org/abs/2512.17399 |