Discrete Double-Bracket Flows for Isotropic-Noise Invariant Eigendecomposition

We study eigendecomposition on SO(n) under streaming observations Ck = Csig + σk2 I + Ek, where the isotropic background σk2 I may be time-varying and arbitrarily large. Standard algorithms couple their stability to lVert Ck rVert2 ≈ σ2, forcing step sizes, contraction rates, and iteration counts to degrade with the noise floor. We observe that σ2 I lies in the center of the matrix algebra and therefore *should never enter* the eigenspace dynamics. We construct a discrete double-bracket flow whose skew-symmetric generator Omega = [A, operatornamediag(A)] operates in the tangent Lie algebra mathfrakso(n), where scalar multiples of the identity vanish by antisymmetry. The resulting trajectory, Lyapunov function, and maximal stable step size ηmax = 1/LC depend exclusively on the trace-free signal Ce -- achieving pointwise, pathwise σ2-invariance. We establish input-to-state stability with a noise ball governed solely by trace-free perturbations, prove global convergence via strict-saddle geometry and a discrete Lojasiewicz argument, and extend the framework to top-k eigentracking on the Stiefel manifold operatornameSt(k,n) at cost k matrix-vector products per step.