Numerical ranges of weighted shift matrices with periodic weights (Q549289)

Let \(A\) be an \(n\)-by-\(n\) (\(n\geq 2\)) matrix with zeros on the main diagonal, \(a_1,\dots, a_{n-1}\) filling the superdiagonal, and with \(a_n\)~in the left bottom position. It is shown that if all \(a_j\)'s are nonzero and their moduli are periodic, then the boundary of its numerical range contains a~line segment. The boundary \(\partial W(A)\) of the numerical range is shown to contain a~noncircular elliptic arc if and only if all \(a_j\)'s are nonzero, \(n\)~is even, \(|a_1|=|a_3|=\dotsc =|a_{n-1}|\), \(|a_2|=|a_4|=\dotsc =|a_n|\) and \(|a_1|\neq |a_2|\). Finally, a~criterion for~\(A\) to be reducible is given, and a complete characterization of the numerical ranges of such matrices is obtained.