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

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.