Spectra of stochastic matrices with a Toeplitz interior
Abstract
For stochastic matrices of order n, the Karpelevič region is the set of all possible eigenvalues. We study matrices whose first and last m rows are arbitrary probability vectors, while the remaining middle rows come from one probability vector by successive cyclic shifts to the right, moving the last entry to the first position each time. As m increases, their eigenvalue regions expand from the polygon spanned by the nth roots of unity. For n≥3, they first equal the full Karpelevič region at m=⌈(n−1)/4⌉. Reproducing all n eigenvalues, counted with multiplicity, of every stochastic matrix of the same order requires m=⌊ n/2⌋. Thus, in even order n≥4, retaining two consecutive rows with the Toeplitz pattern can exclude eigenvalue lists even when every possible eigenvalue occurs individually. The boundary of the Karpelevič region consists of arcs joining roots of unity. We determine the least m for which every point of each arc occurs as an eigenvalue in our family. For arcs whose nonreal points first become possible eigenvalues at order n≥4, and for m≥1 below this minimum, no point on the arc other than its endpoints occurs in the family. At the minimum, every point on the arc becomes attainable. At the smallest order n≥4 at which a nonreal point on a boundary arc away from its endpoints can occur as an eigenvalue, that point determines the characteristic polynomial of every stochastic matrix of order n having it as an eigenvalue. Further results describe the smaller regions and the angles at which their boundaries approach roots of unity.
Related
- Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols
- Revisiting Toeplitz and Hankel random matrices via *-convergence of circulant-type matrices
- Non-Bloch band theory for finite banded non-Hermitian Toeplitz matrices
- Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices
- Constructive recurrences for determinants and permanents of banded Toeplitz matrices
BibTeX
@misc{verbeken2026spectra,
title = {{Spectra of stochastic matrices with a Toeplitz interior}},
author = {Brecht Verbeken and Vincent Ginis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.42072},
eprint = {2610.42072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.42072},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectra of stochastic matrices with a Toeplitz interior
AU - Brecht Verbeken
AU - Vincent Ginis
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.42072
UR - https://arxiv.org/abs/2610.42072
AB - For stochastic matrices of order n, the Karpelevič region is the set of all possible eigenvalues. We study matrices whose first and last m rows are arbitrary probability vectors, while the remaining middle rows come from one probability vector by successive cyclic shifts to the right, moving the last entry to the first position each time. As m increases, their eigenvalue regions expand from the polygon spanned by the nth roots of unity. For n≥3, they first equal the full Karpelevič region at m=⌈(n−1)/4⌉. Reproducing all n eigenvalues, counted with multiplicity, of every stochastic matrix of the same order requires m=⌊ n/2⌋. Thus, in even order n≥4, retaining two consecutive rows with the Toeplitz pattern can exclude eigenvalue lists even when every possible eigenvalue occurs individually. The boundary of the Karpelevič region consists of arcs joining roots of unity. We determine the least m for which every point of each arc occurs as an eigenvalue in our family. For arcs whose nonreal points first become possible eigenvalues at order n≥4, and for m≥1 below this minimum, no point on the arc other than its endpoints occurs in the family. At the minimum, every point on the arc becomes attainable. At the smallest order n≥4 at which a nonreal point on a boundary arc away from its endpoints can occur as an eigenvalue, that point determines the characteristic polynomial of every stochastic matrix of order n having it as an eigenvalue. Further results describe the smaller regions and the angles at which their boundaries approach roots of unity.
ER -