AI 中文总结
本文针对2004年《数学系统与控制理论未解决问题集》中的秩1李雅普诺夫谱恒等式，通过核分解与多项式延拓完成证明，还推导了无稳定性假设的行列式推论及显式相似变换，可恢复相关主角与典型相关谱。
AI 中文摘要
我们证明了2004年《数学系统与控制理论未解决问题集》中问题9.1所记载的秩1李雅普诺夫谱恒等式。设P、Q、R为与矩阵A及其秩1更新A₂=A+vwᵀ相关的耦合离散李雅普诺夫方程和Sylvester方程的解，当显示的逆存在时，我们证明P⁻¹RQ⁻¹Rᵀ与(I+PQ)具有相同的特征多项式。对Q的方程进行秩1行列式分解得到一个标量双线性核，将其在A的特征值及其倒数处求值，可得RQ⁻¹Rᵀ=BQ⁻¹B=P−BPB，之后两个目标矩阵是相同的两个因子按相反顺序排列。多项式延拓将该恒等式从非空的可容许系统开集扩展到整个可容许域，并得到无稳定性假设的行列式推论；当A和A₂的谱不相交时，Z=b(A)⁻¹P给出显式相似变换。在Schur稳定实现场景中，该结果可恢复相关的主角谱及过去/未来典型相关谱。
英文摘要
We prove the rank-one Lyapunov spectral identity recorded as Problem 9.1 in the 2004 collection of unsolved problems in mathematical systems and control theory. Let $P,Q,R$ solve the coupled discrete Lyapunov and Sylvester equations associated with $A$ and its rank-one update $A_2=A+vw^\top$. When the displayed inverses exist, we show that $P^{-1}RQ^{-1}R^\top$ and $(I+PQ)^{-1}$ have the same characteristic polynomial. A rank-one determinant factorization of the equation for $Q$ produces a scalar bilinear kernel. Evaluating it at the eigenvalues of $A$ and at their reciprocals gives $RQ^{-1}R^\top=BQ^{-1}B=P-BPB$, after which the two target matrices are the same two factors in opposite order. Polynomial continuation extends the identity from a nonempty open set of admissible systems to the full admissible domain and yields a determinant corollary without stability assumptions; when the spectra of $A$ and $A_2$ are disjoint, $Z=b(A)^{-1}P$ gives an explicit similarity. In the Schur-stable realization setting, the result recovers the associated principal-angle and past/future canonical-correlation spectra.
Comments13 pages, 3 figures. Submitted to Automatica
