Numerical ranges of companion matrices

Abstract

We show that an n-by-n companion matrix A can have at most n line segments on the boundary a(A) of its numerical range W(A), and it has exactly n line segments on W(A) if and only if, for n odd, A is unitary, and, for n even, A is unitarily equivalent to the direct sum A(1) circle plus A(2) of two (n/2)-by-(n/2) companion matrices [GRAPHICS] with 1 <= vertical bar u vertical bar < tan(pi/n) + sec(pi/n). (c) 2006 Elsevier Inc. All rights reserved.