Diagonals and numerical ranges of direct sums of matrices

Abstract

For any n-by-n matrix A, we consider the maximum number k = k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary partial derivative W(A) of the numerical range W (A) of A. If A is a normal or a quadratic matrix, then the exact value of k(A) can be computed. For a matrix A of the form B circle plus C, we show that k(A) = 2 if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C and k(C) = 2. For an irreducible matrix A, we can determine exactly when the value of k(A) equals the size of A. These are then applied to determine k(A) for a reducible matrix A of size 4 in terms of the shape of W (A). (C) 2013 Elsevier Inc. All rights reserved.