The De Cock--De Moor Conjecture: Proof, Normal Form, and Indefinite Angles

Abstract

De Cock and De Moor conjectured that two matrices built from the solutions of three coupled Lyapunov equations always share their eigenvalues, an identity linking principal angles between model subspaces with the canonical correlations between the past and future of a stochastic process. The conjecture was recorded as Problem 9.1 of \emph{Unsolved Problems in Mathematical Systems and Control Theory}. We prove it under the natural nonresonance condition, without stability or diagonalizability, and show that the two matrices are similar. The proof exposes a Cauchy-matrix structure created by the rank-one perturbation and reduces the problem to rational interpolation, and the main theorem has also been verified in the Lean~4 proof assistant. The same structure gives a normal form that depends only on the poles and zeros of the model. Off the stable set the eigenvalues become squared cosines of angles in an indefinite metric. We prove an inertia theorem for them and show that reflecting a pole or a zero across the unit circle acts on the angles by a principal pivot transform. For several inputs and outputs, where the conjecture fails, we identify the subspaces whose angles do give the canonical correlations. We also relate our approach to the independent proof of Gillberg and Löfberg.