The Karpelevič region revisited

Abstract

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