The Karpelevič Region Revisited

We consider the Karpelevič region $Θ_n \subset \mathbb{C}$ 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 Karpelevič. In particular, for each $θ\in [0, 2π),$ we identify the point on the boundary of $Θ_n$ with argument $θ.$ We further prove that if $n \in \mathbb{N}$ with $n \ge 2,$ and $t \in Θ_n,$ then $t$ is a subdominant eigenvalue of some stochastic matrix of order $n$.

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