LIMITED APPROXIMATION OF NUMERICAL RANGE OF NORMAL MATRIX

Abstract

Let A be an n x n normal matrix, whose numerical range NR[A] is a k-polygon. If a unit vector v is an element of W subset of C(n), with dimW = k and the point v*Av is an element of IntNR[A], then NR[A] is circumscribed to NR[P*AP], where P is an n x (k-1) isometry of {span{v}}(W)(perpendicular to) -> C(n), [1]. In this paper, we investigate an internal approximation of NR[ A] by an increasing sequence of NR[C(s)] of compressed matrices C(s) = R(s)*AR(s), with R(s)*R(s) = I(k+s-1), s = 1,2,..., n - k and additionally NR[A] is expressed as limit of numerical ranges of k-compressions of A.