Numerical ranges of row stochastic matrices

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+n)/2, and, moreover, w(A)=1 (resp., w(A)=(1+n)/2) if and only if A is doubly stochastic (resp.,A=[01⋯10]jth 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.