Refinements for numerical ranges of weighted shift matrices

An n-by-n (n ≥ 3) weighted shift matrix A is one of the form where the a j 's, called the weights of A, are complex numbers. Let A [j] denote the (n - 1)-by-(n - 1) principal submatrix of A obtained by deleting its jth row and jth column. We show that the boundary of numerical range W(A) has a line segment if and only if the a j 's are nonzero andW(A[k]) = W(A[l]) = W(A[m]) for some 1 ≤ k < l < m ≤ n. This refines previous results of Tsai and Wu on numerical ranges of weighted shift matrices. In addition, we give an example showing that there is a weighted shift matrix with line segments on the boundary of its numerical range such that the moduli of its weights are not periodic.