Symmetric maximizers for Lyapunov operators of order six
For $A \in \mathbb{R}^{n \times n}$ let $L_A(X) = AX + XA^{\mathsf T}$ be the Lyapunov operator on $\mathbb{R}^{n \times n}$, and let $\|X\|_F = (\operatorname{tr}(X^{\mathsf T} X))^{1/2}$ be the Frobenius norm. The largest singular value of $L_A$ is the maximum of $\|L_A(X)\|_F$ over $\|X\|_F = 1$. The symmetric and the skew-symmetric matrices form orthogonal invariant subspaces of $L_A$, so this maximum is the larger of the maxima over the two subspaces. Byers and Nash stated that for every $n$ it is attained at a symmetric matrix; Chen and Tian found an error in the proof and proved the statement for $n \le 5$, and Kressner and Vandereycken gave counterexamples for every $n \ge 7$.
Problem. Is $$\max_{X \in \mathbb{R}^{6 \times 6},\ \|X\|_F = 1} \|L_A(X)\|_F = \max_{X \in \mathbb{R}^{6 \times 6},\ X = X^{\mathsf T},\ \|X\|_F = 1} \|L_A(X)\|_F$$ for every $A \in \mathbb{R}^{6 \times 6}$?
Source: Byers and Nash (1987), Theorem 9, whose proof Chen and Tian (2015) found in error; conjectured explicitly by Cheng (2001); order six left open by Kressner and Vandereycken (2026).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- RA
- Keywords
- Lyapunov operators, OpenProblemsInNLA, singular values
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@article{byers1987,
author = {Byers, Ralph and Nash, Stephen},
title = {On the singular ``vectors'' of the {L}yapunov operator},
journal = {SIAM J. Algebraic Discrete Methods},
volume = {8},
number = {1},
year = {1987},
pages = {59--66},
doi = {10.1137/0608003},
}
@article{cheng2001,
author = {Cheng, Daizhan},
title = {On {L}yapunov mapping and its applications},
journal = {Commun. Inf. Syst.},
volume = {1},
number = {3},
year = {2001},
pages = {255--272},
doi = {10.4310/CIS.2001.v1.n3.a2},
}
@article{chen2015,
author = {Chen, Sheng and Tian, Yunbo},
title = {Note on ``{O}n the singular `vectors' of the {L}yapunov operator'' by {R}. {B}yers and {S}. {N}ash},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {36},
number = {3},
year = {2015},
pages = {1069--1072},
doi = {10.1137/140974031},
}
@misc{kressner2026,
author = {Kressner, Daniel and Vandereycken, Bart},
title = {A counterexample to the symmetric-maximizer conjecture for {L}yapunov operators},
year = {2026},
eprint = {2608.20875},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mf13,
author = {Townsend, Alex},
title = {{MF-13}: Symmetric maximizers for Lyapunov operators of order six},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-13/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-00165, "Symmetric maximizers for Lyapunov operators of order six", clopen.solutions.