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arXiv · 2608.20875

A counterexample to the symmetric-maximizer conjecture for Lyapunov operators

Daniel Kressner·Bart Vandereycken
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Abstract

It has been conjectured that the operator norm of the Lyapunov operator induced by the Frobenius norm is always attained at a symmetric matrix. The conjecture is known to hold for all matrices of order at most five. We give an integer matrix of order seven for which the skew-symmetric restricted norm is strictly larger than the symmetric restricted norm. A rational separator and exact-arithmetic certificates establish the strict inequality without relying on floating-point computations. A direct-sum construction yields counterexamples in every order $n \geq 7$; the case $n = 6$ remains open.

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Daniel Kressner, Bart Vandereycken. 2026-08-21. A counterexample to the symmetric-maximizer conjecture for Lyapunov operators. https://arxiv.org/abs/2608.20875

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