Algebraicity of joint spectral radii from rational input
Let $\widehat\rho(\mathcal M)$ be the joint spectral radius of a finite set $\mathcal M$ of real square matrices and $\rho$ the spectral radius. If $\mathcal M$ consists of rational matrices and $\widehat\rho(\mathcal M) = \rho(A_k \cdots A_1)^{1/k}$ for some $k \ge 1$ and $A_1, \dots, A_k \in \mathcal M$ (the finiteness property), then $\widehat\rho(\mathcal M)$ is algebraic, as Jungers notes. Jungers and Blondel showed that every finite set of rational matrices has the finiteness property if and only if every pair of matrices with entries in $\{-1, 0, 1\}$ does.
Problem. Is $\widehat\rho(\mathcal M)$ algebraic, that is, $p(\widehat\rho(\mathcal M)) = 0$ for some nonzero $p \in \mathbb{Z}[t]$, for every $d \ge 1$ and every finite nonempty $\mathcal M \subset \mathbb{Q}^{d \times d}$? Only algebraicity is asked for, not a uniform procedure that finds the minimal polynomial.
Source: Jungers (2008), PhD thesis, §4.4, Open Question 7, its first part; repeated in Jungers (2009), §4.4, Open Question 7.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- NA, OC
- Keywords
- algebraic numbers, joint spectral radius, OpenProblemsInNLA
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@phdthesis{jungers2008thesis,
author = {Jungers, Rapha{\"e}l M.},
title = {Infinite Matrix Products: From the Joint Spectral Radius to Combinatorics},
school = {Universit{\'e} catholique de Louvain},
address = {Louvain-la-Neuve},
year = {2008},
url = {https://research.dial.uclouvain.be/handle/2078.5/88604},
}
@book{jungers2009,
author = {Jungers, Rapha{\"e}l},
title = {The Joint Spectral Radius: Theory and Applications},
series = {Lecture Notes in Control and Information Sciences},
volume = {385},
publisher = {Springer},
address = {Berlin},
year = {2009},
doi = {10.1007/978-3-540-95980-9},
}
@article{jungers2008,
author = {Jungers, Rapha{\"e}l M. and Blondel, Vincent D.},
title = {On the finiteness property for rational matrices},
journal = {Linear Algebra Appl.},
volume = {428},
number = {10},
year = {2008},
pages = {2283--2295},
doi = {10.1016/j.laa.2007.07.007},
eprint = {math/0702489},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mf10,
author = {Townsend, Alex},
title = {{MF-10}: Algebraicity of joint spectral radii from rational input},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-10/README.md},
note = {Read at commit b8c7852, 2026-10-07},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00163, "Algebraicity of joint spectral radii from rational input", clopen.solutions.