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)=1w(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≤n1≤j≤n). A complete characterization of the A  's for which the zero matrix of size n−1n−1 can be dilated to A   is also given. Finally, for each n≥2n≥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.