Numerical ranges of weighted shift matrices with periodic weights

Let A be an n-by-n (n >= 2) matrix of the form [0 a(1) 0 a(n-1) a(n) 0] We show that if the a(j)'s are nonzero and their moduli are periodic, then the boundary of its numerical range contains a line segment. We also prove that partial derivative W (A) contains a noncircular elliptic arc if and only if the a(j)'s are nonzero, n is even, vertical bar a(1)vertical bar = vertical bar a(3)vertical bar = ... = vertical bar a(n-1)vertical bar, vertical bar a(2)vertical bar = vertical bar a(4)vertical bar = ... = vertical bar a(n)vertical bar and vertical bar a(1)vertical bar not equal vertical bar a(2)vertical bar. Finally, we give a criterion for A to be reducible and completely characterize the numerical ranges of such matrices. (C) 2011 Elsevier Inc. All rights reserved.