Numerical range of a normal compression II

Abstract

As in the predecessor [Numerical range of a normal compression, Linear and Multilinear Algebra, in press] of this paper, we consider properties of matrices of the form V*NV, where N = diag(a(1),..., a(n+1)) is a diagonal matrix with distinct eigenvalues a(j)s such that each of them is a corner of the convex hull they generate, and V is an (n + 1)-by-n matrix with V*V = I-n such that any nonzero vector orthogonal to the range space of V has all its components nonzero. We obtain that such a matrix A is determined by its eigenvalues up to unitary equivalence, is irreducible and cyclic, and the boundary of its numerical range is a differentiable curve which contains no line segment. We also consider the condition for the existence of another matrix of the above type which dilates to A such that their numerical ranges share some common points with the boundary of the (n + 1)-gon a(1) (...) a(n+1). (C) 2004 Elsevier Inc. All rights reserved.