Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Verbeken, Brecht F., Ginis, Vincent
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214125832