Numerical ranges of row stochastic matrices

On the numerical range of Kac-Sylvester matrices

“…We remark that 1 n−1 K n is a positive matrix whose columns sum to 1, that is, it is column-stochastic. Having in mind the positivity of the matrix K n , by [8,Proposition 3.3], the following result holds.…”

“…for every θ ∈ [0, 2π), we may conclude that W (K 7 ) is contained in the disc, bounded by the ellipse, centered at the origin, with horizontal major and vertical minor semi-axes of lengths (11), respectively, that is, with the Cartesian equation (8).…”