Abstract
We study the numerical range of an n×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≤6. We conjecture that the maximizing pattern extends to general n×n cyclic shift matrices. For n≤5, we also determine permutations such that the corresponding cyclic shift matrix yields the smallest numerical range.
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