On numerical ranges of the compressions of normal matrices

For an n × n normal matrix A, whose numerical range NRA] is a k-polygon (k ? n), an n × (k − 1) isometry matrix P is constructed by a unit vector υ∈Cⁿ, and NRP∗AP] is inscribed to NRA]. In this paper, using the notations of NRP∗AP] and some properties from projective geometry, an n × n diagonal matrix B and an n × (k − 2) isometry matrix Q are proposed such that NRP∗AP] and NRQ∗BQ] have as common support lines the edges of the k-polygon and share the same boundary points with the polygon. It is proved that the boundary of NRP∗AP] is a differentiable curve and the boundary of the numerical range of a 3 × 3 matrix P∗AP is an ellipse, when the polygon is a quadrilateral.