Numerical radius of Hadamard product of matrices

Abstract

It is known that the numerical radius of the Hadamard product A circle B of two n-by-n matrices A and B is related to those of A and B by (a) w(A circle B) <= 2w(A)w(B), (b) w(A circle B) <= w(A)w(B) if one of A and B is normal, and (c) w(A circle B) <= (max(i) a(ii))w(B) if A = [a(ij)](i,j=1)(n) is positive semidefinite. In this paper, we give complete characterizations of A and B for which the equality is attained. The matrices involved can be considered as elaborate generalizations of the equality-attaining [GRAPHICS] and [GRAPHICS] for (a), [GRAPHICS] (vertical bar a(1)vertical bar >=vertical bar a(2)vertical bar) and [GRAPHICS] for (b), and [GRAPHICS] >= 0 (a(1) >= a(4)) and [GRAPHICS] for (c). (C) 2016 Elsevier Inc. All rights reserved.