A kernel proof of the De Cock-De Moor Lyapunov identity

Abstract

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.