AI 中文总结
本文借助人工智能证明了对于 Hurwitz 伴随矩阵，连续时间 Lyapunov 方程的解在任意半正定强制下逐元素非负，并指出离散时间对应猜想不成立。
AI 中文摘要
我们借助人工智能证明，对于每个实对称强制矩阵 $Q\succeq0$，当状态矩阵为实 Hurwitz 伴随矩阵时，连续时间 Lyapunov 方程的唯一解逐元素非负。这证明了一个早期猜想。该证明不对状态矩阵的谱作任何假设，并通过使用 Horner 多项式矩阵得以实现。我们还表明，该猜想的离散时间对应形式是错误的：可以轻易构造反例，表明 Schur 伴随矩阵（包括由所有系数均为正的多项式生成的矩阵）即使在单位强制下也会产生具有负非对角元素的解。
英文摘要
We prove, with the aid of AI, that for every real symmetric forcing matrix $Q\succeq0$, the unique solution of a continuous-time Lyapunov equation is entrywise nonnegative whenever the state matrix is a real Hurwitz companion matrix. This proves an earlier conjecture. The proof makes no assumption on the spectrum of the state matrix, and is made possible by means of using Horner polynomial matrices. We also show that discrete-time counterpart of the conjecture is false: Counterexamples can be readily constructed to show that Schur companion matrices, including one generated by a polynomial with all positive coefficients, yield solutions with negative off-diagonal entries even with identity forcing.
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