Numerical ranges as circular discs

We prove that if a finite matrix A of the form [ a I B 0 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 ∂ W ( A ) contains a circular arc, then the center of this circle is an eigenvalue of A with its geometric multiplicity strictly less than its algebraic multiplicity, and (b) if A is similar to a normal matrix, then ∂ W ( A ) contains no circular arc.