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 $\widehat{A}$ and a full-rank constraint matrix $C$ encoding one or more invariants, we show that the matrix closest to $\widehat{A}$ in the Frobenius norm and satisfying $C^{\mathrm{\top }}A=0$ is the orthogonal projection $A^{\star }=\widehat{A}-C(C^{\mathrm{\top }}C{)}^{-1}{C}^{\mathrm{\top }}\widehat{A}$. This correction is uniquely defined, low rank and fully determined by the violation $C^{\mathrm{\top }}\widehat{A}$. 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.