Proof of a Conjecture of De Cock and De Moor

Abstract

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.