Numerical ranges and Gersgorin discs

Abstract

For a complex matrix A = vertical bar a(ij)vertical bar(n)(i,j=1), let W(A) be its numerical range, and let G(A) be the convex hull of U-i=1(n) {z is an element of C : vertical bar z - a(ij)vertical bar <= (Sigma(i not equal j)(vertical bar a(ij)vertical bar)/2} and G'(A) = n{G(U*AU) : U n-by-n unitary). It is known that W(A) is always contained in G(A) and hence in G'(A). In this paper, we consider conditions for W (A) to be equal to G(A) or G'(A). We show that if W(A) = G'(A). then the boundary of W(A) consists only of circular arcs and line segments. If, moreover, A is unitarily irreducible, then W (A) is a circular disc. (Almost) complete characterizations of 2-by-2 and 3-by-3 matrices A for which W(A) = G'(A) are obtained. We also give criteria for the equality of W(A) and G(A). In particular, such A's among the permutationally irreducible ones must have even sizes. We also characterize those A's with size 2 or 4 which satisfy W(A) = G(A). Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.