Numerical ranges as circular discs

AbstractWe prove that if a finite matrix A of the form [aIB0C]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