Numerical ranges of companion matrices
The authors show that an \(n\)-by-\(n\) companion matrix \(A\) can have at most \(n\) line segments on the boundary \(\partial W(A)\) of its numerical range \(W(A)\), and it has exactly \(n\) line segments on \(\partial W(A)\) if and only if, for \(n\) odd, \(A\) is unitary, and, for \(n\) even, \(A\) is unitarily equivalent to the direct sum \(A_1\oplus A_2\) of two \((n/2)\)-by-\((n/2)\) companion matrices \(A_1\) and \(A_2\) with \(1\leqslant | A| <\tan (\pi/n) + \sec (\pi/n)\). Here for both \(A_1\) and \(A_2\) the main diagonal is totally 0, their superdiagonal is totally 1, and the lower-left entry of \(A_1\) is \(a\), and \(-1/\bar a\), for \(A_2\).
scientific article; zbMATH DE number 5132591