= Stability and normalization of averaged Oja learning
{title2=$\tau\dot u_j=(\lambda_j-\lambda_i)u_j,\quad\tau\dot u_i=-2\lambda_i u_i$}
At the equilibrium $v_i/\sqrt\alpha$, linearization along orthonormal eigenvectors of $C$ gives the displayed rates, with the first equation applying to $j\ne i$. A simple positive largest eigenvalue makes its two normalized eigenvectors locally asymptotically stable; smaller eigenvalues have an unstable direction. Repeated largest eigenvalues give a manifold of equilibria with neutral directions. Generic convergence also needs a nonzero initial component in the largest eigenspace. The squared norm satisfies $\tau\,d\|w\|^2/dt=2(w^TCw)(1-\alpha\|w\|^2)$.
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