Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices§

Abstract

We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A 1,A 2, an n × n nilpotent Toeplitz matrix Nn , and an n × n cyclic permutation matrix Sn (s) such that the numbers of flat portions on the boundaries of W(A 1⊕N n ) and W(A 2⊕S n (s)) are, respectively, 2(n − 2) and 2n. § This article was presented at the 8th Workshop on Numerical Ranges and Numerical Radii, July 15–17, 2006, Universität Bremen, Germany.