Numerical ranges of weighted shift matrices with periodic weights

Let A be an n-by-n   (n⩾2n⩾2) matrix of the form0a10⋱⋱an-1an0.We show that if the ajaj’s are nonzero and their moduli are periodic, then the boundary of its numerical range contains a line segment. We also prove that ∂W(A)∂W(A) contains a noncircular elliptic arc if and only if the ajaj’s are nonzero, n   is even, |a1|=|a3|=⋯=|an-1||a1|=|a3|=⋯=|an-1|, |a2|=|a4|=⋯=|an||a2|=|a4|=⋯=|an| and |a1|≠|a2||a1|≠|a2|. Finally, we give a criterion for A to be reducible and completely characterize the numerical ranges of such matrices.