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Proof of a Conjecture of De Cock and De Moor

Jeffrey Humpherys

arXiv:2608.29479v1math.OCeess.SY

TL;DR

The paper proves the De Cock–De Moor conjecture connecting Lyapunov-equation matrices with principal-angle and canonical-correlation viewpoints in stochastic subspace identification. It uses rank-one perturbation structure to reveal Cauchy matrices and reduce the generic case to rational interpolation, then applies density and continuity. The result establishes cospectrality under nonresonance, automatic nonsingularity of R and I + PQ, and similarity on the generic set and, via the independent intertwiner, throughout the theorem’s domain.

  • Problem

    The paper addresses the conjecture that two differently formed matrices from Stein/Lyapunov solutions and the product PQ share eigenvalues.

  • Method

    The proof combines rank-one eigenbasis factorization, Cauchy-matrix representations of Stein solutions, rational interpolation, and density-continuity arguments.

  • Results

    Under nonresonance, the matrices have the same characteristic polynomial with algebraic multiplicity; R and I + PQ are automatically nonsingular, and the matrices are similar on the generic set.

  • Takeaways & Limitations

    The spectral identity is an algebraic consequence of the interaction among rank-one perturbations, Stein equations, and Cauchy structure, without stability or diagonalizability assumptions.

  • Takeaways & Limitations

    The rank-one scalar factorization does not extend directly to higher-rank perturbations, for which unrestricted multicolumn analogues have counterexamples.

Abstract

from arXiv · show

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.

1 Introduction

The paper addresses the De Cock–De Moor conjecture linking Lyapunov-equation matrices with principal angles and canonical correlations in stochastic subspace identification. It develops a finite-dimensional proof using rank-one perturbations, Cauchy structure, rational interpolation, and density, without stability or diagonalizability assumptions.

  • Motivation: The conjecture connects Lyapunov-equation expressions with the geometry and statistical dependence of past and future model subspaces.Principal angles measure subspace separation, while canonical correlations measure dependence and, for Gaussian processes, determine mutual information.
  • The conjecture: The conjectured matrices have the same eigenvalues despite being formed from different combinations of P, Q, R, and PQ.One matrix uses the cross solution R together with P and Q, whereas the other depends only on PQ.
  • Proof strategy: The proof diagonalizes A and A+vw^T generically, exposes Cauchy matrices in the Stein solutions, and uses rank-one eigenbasis factorization with rational interpolation.The two target matrices become similar to the same explicitly constructed matrix on the generic set.
  • Proof strategy: Density and continuity remove the auxiliary generic spectral assumptions and extend the characteristic-polynomial identity to arbitrary nonresonant data.Continuity of the Stein solutions and limiting characteristic polynomials supplies the nongeneric conclusion.
  • Contributions: The result strengthens the original formulation: nonsingularity of I + PQ becomes automatic, R is automatically nonsingular, and generic similarity replaces mere cospectrality.The argument is finite-dimensional and does not require Schur stability, Neumann-series representations, or diagonalizability.

2 The De Cock–De Moor conjecture

Under nonresonance, the Stein equations uniquely define P, Q, and R, and the conjecture holds whenever P and Q are nonsingular. The conclusion includes automatic nonsingularity of R and I + PQ, equality of eigenvalues with algebraic multiplicity, and generic similarity.

  • Hypotheses: Nonresonance excludes products of eigenvalues of A and B equal to 1, ensuring unique solvability of the three Stein equations.The vectorized coefficient matrices have eigenvalues 1−α_iα_j, 1−β_iβ_j, and 1−α_iβ_j.
  • Theorem: The formulation removes the separate assumption that I + PQ be nonsingular.This nonsingularity is derived rather than imposed in the theorem.
  • Theorem: If P and Q are nonsingular, then R and I + PQ are also nonsingular under nonresonance.These are automatic consequences of the theorem’s hypotheses.
  • Theorem: The two target matrices have identical eigenvalues counted with algebraic multiplicity, equivalently the same characteristic polynomial.When A and B have simple, nonzero spectra, the matrices are additionally similar.

3 Cauchy structure on the simple-spectrum set

On the simple-spectrum set, diagonalizing A and B transforms the Stein solutions into diagonally scaled Cauchy matrices. The rank-one perturbation makes the eigenbasis change another scaled Cauchy matrix, enabling a common similarity representative.

  • Generic setup: The generic assumptions require A and B to have distinct, nonzero eigenvalues and P and Q to be nonsingular.Complex diagonalizations are used, with plain transposes and bilinear identities rather than conjugate transposes.
  • Generic setup: Two-sided eigenbasis transformations reduce the Stein equations to diagonal equations involving the eigenvalue matrices and transformed vectors.The transformed solutions are bP, bQ, and bR, while x and y encode v and w in the two eigenbases.
  • Cauchy structure: The transformed Stein solutions factor as bP = D_xG_AD_x, bQ = D_yG_BD_y, and bR = D_xHD_y, where G_A, G_B, and H are Cauchy matrices.Nonresonance makes the denominators nonzero, and the generalized Cauchy determinant gives nonsingularity of the Cauchy factors.
  • Cauchy structure: The rank-one relation B−A = vw^T forces the eigenbasis-change matrix to be a diagonally scaled Cauchy matrix.It also implies disjoint spectra and nonsingularity of the associated Cauchy factor.
  • Similarity conclusion: The Cauchy identities yield similarities between the theorem’s matrices and a common representative, so their characteristic polynomials coincide.The similarities are established over C and transfer to the real matrices because their characteristic polynomials are real.

4 A rational interpolation identity

A rational interpolation identity supplies the Cauchy-matrix relations needed to prove the spectral result. Reciprocal-node identities then connect the interpolation formula to the matrices arising from the Stein equations.

  • Interpolation identity: For distinct nodes with nonresonant products, the functions f_ζ(β_j) = 1/(1−β_jζ) form the basic interpolation data.The proof uses rational functions whose numerator vanishes at the interpolation nodes and therefore is divisible by 1−zζ.
  • Interpolation identity: The interpolation identity follows because the constructed numerator has degree at most n−1 and vanishes at n distinct nodes.This proves the identity first for admissible parameters and then as a rational-function identity.
  • Interpretation: The identity has an algebraic model-space interpretation for real nodes in (−1,1), while the paper proves it directly for general complex nodes by rational interpolation.In that interpretation, G_β is the Gram matrix of Szegő kernels.
  • Applications: Applying the identity to rows of the Cauchy matrix H produces the first Cauchy relation used in the similarity argument.The evaluation points are eigenvalues of A, admissible by nonresonance and spectral disjointness.
  • Applications: Using reciprocal nodes produces a second identity involving G_{b^{-1}} and the change-of-basis factor C.The reciprocal family is admissible because the original nodes are distinct and nonzero and satisfy the nonresonance conditions.

5 Proof on the simple-spectrum set

On the simple-spectrum set, the proof shows that both conjectured matrices are similar to the same explicit representative. Their common characteristic polynomial depends only on the spectra of A and B, yielding a pole–zero symmetry.

  • 5 Proof on the simple-spectrum set: Under (S1)–(S2), the proof establishes nonsingularity of the matrices needed to compare the two conjectured expressions.The argument uses nonsingular factors in the Cauchy-matrix representation.
  • 5 Proof on the simple-spectrum set: Both target matrices are similar to the common representative I − eE, so they have the same characteristic polynomial.This similarity is proved on the simple-spectrum set, where the Cauchy structure is available.
  • 5 Proof on the simple-spectrum set: The common characteristic polynomial depends only on the spectra of A and B, not otherwise on the perturbation vectors v and w.This spectral dependence is symmetric in the two node families.
  • 5 Proof on the simple-spectrum set: Exchanging the spectral families leaves the characteristic polynomial unchanged, expressing pole–zero duality in the stable SISO interpretation.The two matrices arise from products of the same nonsingular factors in opposite order.

6 Removal of the generic assumptions

The generic spectral assumptions are removed by approximating admissible data with simple, nonzero spectra and passing the identities to the limit by continuity. A determinant identity simultaneously makes R and I + PQ nonsingular automatic.

  • 6 Removal of the generic assumptions: Density of simple, nonzero spectra lets every admissible parameter triple be approximated by triples satisfying the generic assumptions.The admissible set is open because the Stein solutions depend continuously on the data.
  • 6 Removal of the generic assumptions: The failure of simple, nonzero spectra is contained in the zero set of d(A,v,w) = det(A) det(B) disc(pA) disc(pB).Because this polynomial is not identically zero, its nonzero set is dense.
  • 6 Removal of the generic assumptions: det(R)^2 det(I + PQ) = det(P) det(Q), so nonsingular P and Q force both R and I + PQ to be nonsingular.This makes the separate nonsingularity assumption on I + PQ unnecessary.
  • 6 Removal of the generic assumptions: Continuity of Stein solutions, matrix inversion, and characteristic-polynomial coefficients transfers the generic spectral assertion to the full nonresonant domain.The similarity claim on the generic set yields the theorem’s final similarity assertion through Proposition 5.1.
  • 6 Removal of the generic assumptions: The same proof applies to complex A, v, and w using the ordinary transpose.The corresponding generic spectral set is dense over C.

7 Discussion

The discussion isolates rank-one perturbations, Stein equations, and Cauchy structure as the algebraic mechanism behind the conjecture, while clarifying the hypotheses and system-theoretic connection. It also records generic similarity, automatic nonsingularity, and the rank-one limitation on extensions.

  • Algebraic mechanism: Rank-one perturbations force a diagonally scaled Cauchy change-of-basis matrix, while Stein equations produce a second Cauchy family whose spectral identity reduces to rational interpolation.This interaction explains the equality of spectra.
  • Hypotheses: Stability is unnecessary for the finite-dimensional argument; nonresonance is the natural algebraic hypothesis, and density and continuity remove the generic spectral assumptions.The same limiting argument makes I + PQ nonsingular automatic when P and Q are nonsingular.
  • Systems connection: The theorem strengthens the stable systems interpretation by upgrading a product relation involving principal angles and canonical correlations to equality of full characteristic polynomials.In the stable SISO setting, the relevant spectra correspond to transfer-function poles and zeros.
  • Similarity: On the generic set, both target matrices are similar to the common representative I − G−1 A ΦG A Φ, whose characteristic polynomial depends only on the spectra of A and B.The displayed representative is reported across the cited passage fragments.
  • Similarity: The Gillberg–Löfberg intertwiner extends similarity beyond the simple-spectrum set and is nonsingular throughout the theorem's domain.The independently posted proof uses a Lyapunov-kernel identity and the classical AB–BA principle.
  • Scope: Higher-rank extensions do not follow by replacing scalar rank-one factors with blocks, because the scalar factorization is lost and unrestricted multicolumn analogues have counterexamples.Additional structure would be required for a useful higher-rank extension.
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