The Karpelevi\v{c} Region Revisited

We consider the Karpelevi\v{c} region ${\mathrm{\Theta }}_n\subset \mathbb{C}$ consisting of all eigenvalues of all stochastic matrices of order $n$. We provide an alternative characterisation of ${\mathrm{\Theta }}_n$ that sharpens the original description given by Karpelevi\v{c}. In particular, for each $\theta \in [0, 2\pi),$weidentifythepointontheboundaryof$\Theta_n$ with argument $\theta \mathrm{.}$ We further prove that if $n\in \mathbb{N}$ with $n\ge 2,$ and $t\in {\mathrm{\Theta }}_n,$ then $t$ is a subdominant eigenvalue of some stochastic matrix of order $n$.