Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding

Section II of the paper states that for a finite set of independent observables g_i in a Hilbert space H, one can form the inner-product matrices R_{ij} = ⟨g_i, g_j⟩ and Q_{ij} = ⟨g_i ∘ f, g_j⟩, and obtain the finite-dimensional state-transition matrix via A = Q R^{-1} (the paper motivates this after an orthonormal-basis presentation) . The candidate solution explicitly derives Q = A R from the invariance assumption g_i ∘ f ∈ span{g_1,...,g_N} and R invertible, giving A = Q R^{-1}, and then shows χ[f(x)] = A χ(x). This matches the paper’s formula and intent; the model fills in the missing algebraic steps and makes the invariance and invertibility assumptions explicit. Minor differences are in phrasing: the paper uses "complete and independent" language that tacitly entails the needed invertibility/invariance, while the model states them precisely.