A Frobenius-Optimal Projection for Enforcing Linear Conservation in Learned Dynamical Models

We consider the problem of restoring linear conservation laws in data-driven linear dynamical models. Given a learned operator widehatA and a full-rank constraint matrix C encoding one or more invariants, we show that the matrix closest to widehatA in the Frobenius norm and satisfying C^top A = 0 is the orthogonal projection A^star = widehatA - C(C^top C)-1C^top widehatA. This correction is uniquely defined, low rank and fully determined by the violation C^top widehatA. In the single-invariant case it reduces to a rank-one update. We prove that A^star enforces exact conservation while minimally perturbing the dynamics, and we verify these properties numerically on a Markov-type example. The projection provides an elementary and general mechanism for embedding exact invariants into any learned linear model.