The Karpelevič Region Revisited

Abstract

We consider the Karpelevič region ${\Theta }_n\subset \mathbb{C}$ 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 Karpelevič. 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 $n\in \mathbb{N}$ with $n\ge 2,$ and $t\in {\Theta }_n,$ then $t$ is a subdominant eigenvalue of some stochastic matrix of order $n$.

26 pages, 2 Figures