Numerical ranges of row stochastic matrices

Abstract

In this paper, we consider properties of the numerical range of an n-by-n row stochastic matrix A. It is shown that the numerical radius of A satisfies 1 <= w(A) <= (1 + root n)/2, and, moreover, w(A) = 1 (resp., w(A) = (1 + root n)/2) if and only if A is doubly stochastic (resp., A = [GRAPHICS] for some j, 1 <= j <= n). A complete characterization of the A\'s for which the zero matrix of size n - 1 can be dilated to A is also given. Finally, for each n >= 2, we determine the smallest rectangular region in the complex plane whose sides are parallel to the x- and y-axis and which contains the numerical ranges of all n-by-n row stochastic matrices. (C) 2016 Elsevier Inc. All rights reserved.