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.
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BibTeX
@misc{verbeken2026proof,
title = {{A proof of the Karpelevič theorem via invariant polygons}},
author = {Brecht F. Verbeken and Vincent Ginis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.26058},
eprint = {2609.26058},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26058},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A proof of the Karpelevič theorem via invariant polygons
AU - Brecht F. Verbeken
AU - Vincent Ginis
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.26058
UR - https://arxiv.org/abs/2609.26058
AB - 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.
ER -