A proof of the Karpelevič theorem via invariant polygons

Abstract

The Karpelevič theorem describes the complex numbers that occur as eigenvalues of stochastic matrices of a fixed order. We give a self-contained proof based on invariant polygons. Their geometry yields a product equation for an eigenvalue of maximal modulus at a fixed argument. Convexity determines this modulus, explicit stochastic matrices attain it, and a comparison between Farey intervals completes the boundary description. As the matrix order increases, the resulting equation gives a uniform relative asymptotic for the gap $1-|z|$ between a boundary point $z$ and the unit circle, even arbitrarily close to arc endpoints. We also determine the optimal contraction factor for a planar rotation and dilation when vector size is measured by scaling a polygon with at most $N$ vertices.

arXiv comment62 pages, 3 figures. Exposition substantially revised and references corrected; title changed from "A structural proof of the Karpelevič theorem". Includes a self-contained proof via invariant polygons, applications to optimal polygonal contraction factors, and uniform Farey asymptotics