arXiv subjects

Helena Smigoc

Publications and source records attributed to Helena Smigoc.

2 recordsLinked to original sources

arXiv2023

Powers of Karpelevic arcs and their Sparsest Realising matrices

The region in the complex plane containing the eigenvalues of all stochastic matrices of order n was described by Karpelevic in 1988, and it is since then known as the Karpelevic region. The boundary of the Karpelevic region is the union of disjoint arcs called the Karpelevic arcs. We provide a complete characterization of the Karpelevic arcs that are powers of some other Karpelevic arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevic arc of order n to be a power of another stochastic matrix.

math.CO↗

arXiv2020

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 $n$.

math.SP↗