Proofs of conjectures on the Karpelevich arcs in the region of eigenvalues of stochastic matrices

Abstract

The subset of the complex plane that consists of all eigenvalues of all stochastic matrices of a fixed order was completely determined by Karpelevich (1951). The boundary of this region consists of so-called Karpelevich arcs. Johnson and Paparella (2017) [6] made several conjectures on properties of these arcs. In this paper, we prove two of their conjectures. Specifically, we prove that the Karpelevich arcs are regular differentiable curves and establish that some powers of certain Karpelevich arcs correspond to some other Karpelevich arcs. (C) 2020 Elsevier Inc. All rights reserved.