Numerical ranges as circular discs

Abstract

We prove that if a finite matrix A of the form [(al)(0) (B)(C)]is such that its numerical range W (A) is a circular disc centered at a, then a must be an eigenvalue of C. As consequences, we obtain, for any finite matrix A, that (a) if aW (A) contains a circular arc, then the center of this circle is an eigenvalue ofA with its geometric multiplicity strictly less than its algebraic multiplicity, and (b) if A is similar to a normal matrix, then aW (A) contains no circular arc. (C) 2011 Elsevier Ltd. All rights reserved.