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arXiv · 2304.06050

Numerical ranges of cyclic shift matrices

Mao-Ting Chien·Steve Kirkland·Chi-Kwong Li·Hiroshi Nakazato
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Abstract

We study the numerical range of an $n\times n$ cyclic shift matrix, which can be viewed as the adjacency matrix of a directed cycle with $n$ weighted arcs. In particular, we consider the change in the numerical range if the weights are rearranged or perturbed. In addition to obtaining some general results on the problem, a permutation of the given weights is identified such that the corresponding matrix yields the largest numerical range (in terms of set inclusion), for $n \le 6$. We conjecture that the maximizing pattern extends to general $n\times n$ cylic shift matrices. For $n \le 5$, we also determine permutations such that the corresponding cyclic shift matrix yields the smallest numerical range.

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Mao-Ting Chien, Steve Kirkland, Chi-Kwong Li, Hiroshi Nakazato. 2023-04-12. Numerical ranges of cyclic shift matrices. https://arxiv.org/abs/2304.06050

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