The Karpelevic region revisited

Abstract

We consider the Karpelevic region \Theta_n in the complex plane consisting of all eigenvalues of all stochastic matrices of order n. We provide an alternative characterisation of \Theta_n that sharpens the original description given by Karpelevic. In particular, for each \theta in [0; 2\pi); we identify the point on the boundary of \Theta_n with argument \theta. We further prove that if n is a natural number with n at least 2, and t is in \Theta_n, then t is a subdominant eigenvalue of some stochastic matrix of order n.