De Cock与De Moor猜想的证明

中文摘要 AI 辅助

De Cock和De Moor提出了一个猜想，该猜想联系了随机子空间识别中两个看似不同的观点：一个基于Lyapunov方程，另一个基于主角度和典型相关系数。该猜想被记录在《数学系统与控制理论中的未解决问题》的问题9.1中。我们在自然非共振条件下给出了一个直接的有限维证明，无需稳定性或可对角化条件。关键机制是秩一扰动，它揭示了隐藏的Cauchy矩阵结构，并将问题简化为有理插值。随后通过密度与连续性论证消除了通用谱假设。该结果强化了原命题：特征值与代数重数一致，原公式的非奇异性假设变为自动成立；在参数的一个稠密开集上，两个矩阵相似，而非仅余谱相等。在本文稿准备期间，Gillberg和Löfberg独立发布了基于Lyapunov核恒等式与经典AB-BA原理的证明，本文给出的证明为独立完成，且遵循不同路径。

英文摘要

De Cock and De Moor proposed a conjecture connecting two seemingly different viewpoints in stochastic subspace identification, one based on Lyapunov equations and the other on principal angles and canonical correlations. The conjecture was recorded as Problem 9.1 of \emph{Unsolved Problems in Mathematical Systems and Control Theory}. We give a direct finite-dimensional proof under the natural nonresonance condition, without requiring stability or diagonalizability. The key mechanism is the rank-one perturbation, which exposes a hidden Cauchy-matrix structure and reduces the problem to rational interpolation. A density and continuity argument then removes the generic spectral assumptions. The result strengthens the original statement. The eigenvalues agree with algebraic multiplicity, a nonsingularity assumption of the original formulation becomes automatic, and on a dense open set of parameters the two matrices are similar rather than merely cospectral. While this manuscript was being prepared, Gillberg and Löfberg independently posted a proof based on a Lyapunov-kernel identity and the classical $AB$--$BA$ principle. The proof given here was developed independently and follows a different route.

发表机构

• Missouri University of Science and Technology（密苏里科技大学）

机构由 AI 辅助整理，请以论文原文为准。