Refinements for numerical ranges of weighted shift matrices

An n-by-n () weighted shift matrix A is one of the form where the 's, called the weights of A, are complex numbers. Let denote the -by- 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 's are nonzero and for some . 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.