The Karpelevič region revisited

We consider the Karpelevic region Θ n ⊂ C consisting of all eigenvalues of all stochastic matrices of order n. We provide an alternative characterisation of Θ n that sharpens the original description given by Karpelevic. In particular, for each θ ∈ [ 0 , 2 π ) , we identify the point on the boundary of Θ n with argument θ. We further prove that if n ∈ N with n ≥ 2 , and t ∈ Θ n , then t is a subdominant eigenvalue of some stochastic matrix of order n.