Numerical ranges of weighted shift matrices

We show that if A is an n-by-n   (n⩾3n⩾3) matrix of the form0a10⋱⋱an-1an0,then the boundary of its numerical range contains a line segment if and only if the ajaj’s are nonzero and the numerical ranges of the (n-1n-1)-by-(n-1n-1) principal submatrices of A   are all equal. For n=3n=3, this is the case if and only if |a1|=|a2|=|a3|≠0|a1|=|a2|=|a3|≠0, in which case W(A)W(A), the numerical range of A  , is the equilateral triangular region with vertices the three cubic roots of a1a2a3a1a2a3. For n=4n=4, the condition becomes |a1|=|a3|≠0|a1|=|a3|≠0 and |a2|=|a4|≠0|a2|=|a4|≠0, in which case W(A)W(A) is the convex hull of two (degenerate or otherwise) ellipses.