Finiteness for nonnegative rational matrix families
Let $\widehat\rho(\mathcal M)$ be the joint spectral radius of a finite set $\mathcal M$ of real $d \times d$ matrices and $\rho$ the spectral radius. The set $\mathcal M$ has the finiteness property if $$\widehat\rho(\mathcal M) = \rho(A_k \cdots A_1)^{1/k}$$ for some $k \ge 1$ and $A_1, \dots, A_k \in \mathcal M$. Lagarias and Wang conjectured that every finite set of real matrices has it, which Bousch and Mairesse disproved. Jungers and Blondel showed that every finite set of matrices with nonnegative rational entries has the finiteness property if and only if every pair of matrices with entries in $\{0, 1\}$ does, and that every such pair of $2 \times 2$ matrices does; Mejstrik showed, with computer assistance, that every such pair of $3 \times 3$ matrices does. A positive answer to the question below would make it decidable whether $\widehat\rho(\mathcal M) < 1$ for finite sets $\mathcal M$ of matrices with nonnegative rational entries (Jungers and Blondel).
Problem. For every $d \ge 1$, does every finite nonempty set $\mathcal M$ of $d \times d$ matrices with nonnegative rational entries have the finiteness property, that is, are there $k \ge 1$ and $A_1, \dots, A_k \in \mathcal M$ with $\widehat\rho(\mathcal M) = \rho(A_k \cdots A_1)^{1/k}$?
Source: Jungers and Blondel (2008), §1, who ask whether rational matrices have the finiteness property and show (Theorem 1) that for nonnegative ones this is equivalent to their Conjecture 1, that pairs of binary matrices have it, which they credit to Blondel, Jungers and Protasov (2006); Jungers (2009), §4.4, Open Question 5.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- NA, OC
- Keywords
- finiteness property, joint spectral radius, nonnegative matrices, OpenProblemsInNLA
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@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},
}
@article{blondel2006,
author = {Blondel, Vincent D. and Jungers, Rapha{\"e}l and Protasov, Vladimir},
title = {On the complexity of computing the capacity of codes that avoid forbidden difference patterns},
journal = {IEEE Trans. Inform. Theory},
volume = {52},
number = {11},
year = {2006},
pages = {5122--5127},
doi = {10.1109/TIT.2006.883615},
eprint = {cs/0601036},
archivePrefix = {arXiv},
}
@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{lagarias1995,
author = {Lagarias, Jeffrey C. and Wang, Yang},
title = {The finiteness conjecture for the generalized spectral radius of a set of matrices},
journal = {Linear Algebra Appl.},
volume = {214},
year = {1995},
pages = {17--42},
doi = {10.1016/0024-3795(93)00052-2},
}
@article{bousch2002,
author = {Bousch, Thierry and Mairesse, Jean},
title = {Asymptotic height optimization for topical {IFS}, {T}etris heaps, and the finiteness conjecture},
journal = {J. Amer. Math. Soc.},
volume = {15},
number = {1},
year = {2002},
pages = {77--111},
doi = {10.1090/S0894-0347-01-00378-2},
}
@article{mejstrik2022,
author = {Mejstrik, Thomas},
title = {The finiteness conjecture for $3 \times 3$ binary matrices},
journal = {Dolomites Res. Notes Approx.},
volume = {15},
number = {5},
year = {2022},
pages = {24--38},
doi = {10.14658/PUPJ-DRNA-2022-5-3},
eprint = {2505.10178},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mf04,
author = {Townsend, Alex},
title = {{MF-04}: Finiteness for nonnegative rational matrix families},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-04/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-00161, "Finiteness for nonnegative rational matrix families", clopen.solutions.