A matricial view of the Karpelevi\v{c} Theorem

TL;DR

This paper provides a simplified, matrix-based geometric view of the Karpelevic region, describing the eigenvalues of stochastic matrices and exploring related questions and conjectures.

Contribution

It introduces a matricial parametrization of the boundary arcs of the Karpelevic region, simplifying the understanding of eigenvalue regions for stochastic matrices.

Findings

Each boundary arc is realized by a single parametrized stochastic matrix.

The structure of the arcs and their relation to roots of unity are analyzed.

Open questions and conjectures about the doubly stochastic analog are discussed.

Abstract

The question of the exact region in the complex plane of the possible single eigenvalues of all $n$-by-$n$ stochastic matrices was raised by Kolmogorov in 1937 and settled by Karpelevi\v{c} in 1951 after a partial result by Dmitriev and Dynkin in 1946. The Karpelevi\v{c} result is unwieldy, but a simplification was given by {\DJ}okovi\'c in 1990 and Ito in 1997. The Karpelevi\v{c} region is determined by a set of boundary arcs each connecting consecutive roots of unity of order less than $n$. It is shown here that each of these arcs is realized by a single, somewhat simple, parametrized stochastic matrix. Other observations are made about the nature of the arcs and several further questions are raised. The doubly stochastic analog of the Karpelevi\v{c} region remains open, but a conjecture about it is amplified.