关于Lyapunov算子对称极大值猜想的一个反例
A counterexample to the symmetric-maximizer conjecture for Lyapunov operators
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中文总结 AI 辅助
研究针对Lyapunov算子对称极大值猜想，通过7阶整数矩阵及直和构造找到n≥7阶反例，证明猜想在n≥7时不成立，n=6情况仍未知。
中文摘要 AI 辅助
已有猜想指出，由Frobenius范数诱导的Lyapunov算子的算子范数总是在对称矩阵处取得。该猜想对所有阶数不超过5的矩阵成立。我们给出一个7阶整数矩阵，其斜对称限制范数严格大于对称限制范数。有理分离器和精确算术证书在不依赖浮点计算的情况下证明了该严格不等式。直和构造给出了所有阶数n≥7的反例；n=6的情况仍待解决。
英文摘要
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.
