1 Dilation of numerical ranges of normal matrices

By Maria Adam and John Maroulas.

Let A be an n × n normal matrix, whose the numerical range NR[A] is a k−polygon, and let v ∈ ℂ^{n}, be a unit vector. If for a unit vector v ∈ ℂ^{n}, the point v^{∗}Av is interior point of NR[A] and P is an n×(k −1) matrix, such that P^{∗}P = I_{k−1} and v⊥ImP, then NR[A] is circumscribed to NR[C], where C = P^{∗}AP. In this paper, we investigate the converse way, showing how we obtain NR[A], from a (k − 1)−polygon, such that the boundary of NR[C] shares the same tangential points with the sides of both polygons.