Numerical ranges of weighted shift matrices

Abstract

We show that if A is an n-by-n (n >= 3) matrix of the form (sic) then the boundary of its numerical range contains a line segment if and only if the a(j)'s are nonzero and the numerical ranges of the (n - 1)-by-(n - 1) principal submatrices of A are all equal. For n = 3, this is the case if and only if vertical bar a(1)vertical bar = vertical bar a(2)vertical bar = vertical bar a(3)vertical bar not equal 0, in which case W (A), the numerical range of A, is the equilateral triangular region with vertices the three cubic roots of aia2a3. For n = 4, the condition becomes vertical bar a(1)vertical bar = vertical bar a(3)vertical bar not equal 0 and vertical bar a(2)vertical bar = vertical bar a(4)vertical bar not equal 0, in which case W (A) is the convex hull of two (degenerate or otherwise) ellipses. (C) 2011 Elsevier Inc. All rights reserved.