Zero-dilation index of a finite matrix

Abstract

For an n-by-n complex matrix A, we define its zero-dilation index d(A) as the largest size of a zero matrix which can he dilated to A. This is the same as the maximum k (>= 1) for which 0 is in the rank-k numerical range of A. Using a result of Li and Sze, we show that if d(A) > left perpendicular2n/3right perpendicular, then, under unitary similarity, A has the zero matrix of size 3d(A) - 2n as a direct summand. It complements the known fact that if d(A) > left perpendicularn/2right perpendicular, then 0 is an eigenvalue of A. We then use it to give a complete characterization of n-by-n matrices A with d(A) = n - 1, namely, A satisfies this condition if and only if it is unitarily similar to B circle times 0(n-3), where B is a 3-by-3 matrix whose numerical range W (B) is an elliptic disc and whose eigenvalue other than the two foci of partial derivative W (B) is 0. We also determine the value of d(A) for any normal matrix A and any weighted permutation matrix A with zero diagonals. (C)2013 Elsevier Inc. All rights reserved.