On numerical ranges of the compressions of normal matrices

Abstract

For an n x n normal matrix A, whose numerical range NR[A] is a k-polygon (k <= n), an n x (k - 1) isometry matrix P is constructed by a unit vector upsilon is an element of C(n), and NR[P*AP] is inscribed to NR[A]. In this paper, using the notations of NR[P*AP] and some properties from projective geometry, an n x n diagonal matrix B and an n x (k - 2) isometry matrix Q are proposed such that NR[P*AP] and NR[Q*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 NR[P*AP] is a differentiable curve and the boundary of the numerical range of a 3 x 3 matrix P*AP is an ellipse, when the polygon is a quadrilateral. (C) 2010 Elsevier Inc. All rights reserved.