For n>3, companion matrices with repeated eigenvalue a and circular numerical range must be Jordan blocks; extended to spectra {0,a} with given multiplicities.
Elementary proofs for some results on the circular symmetry of the numerical range.Linear Multilinear Algebra, 61:643–647, 2013
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On Circular Numerical Ranges of Companion Matrices with Repeated Eigenvalues
For n>3, companion matrices with repeated eigenvalue a and circular numerical range must be Jordan blocks; extended to spectra {0,a} with given multiplicities.