3 Numerical Radii for Tensor Products of Matrices
Hwa-Long Gau, Kuo-Zhong Wang, Pei Yuan Wu · 2016
For-by- and-by- complex matrices and, it is known that the inequality holds, where and denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if and, then one of the following two conditions holds: (i) has a unitary part, and (ii) is completely nonunitary and the numerical range of is a circular disc centered at the origin, (2) if for some , , then , and, moreover, the equality holds if and only if is unitarily similar to the direct sum of the -by- Jordan block and a matrix with , and (3) if is a nonnegative matrix with its real part (permutationally) irreducible, then , if and only if either or and is permutationally similar to a block-shift matrixwith , where and .