Stability and normalization of averaged Oja learning ($\tau {\dot{u}}_j=({\lambda }_j-{\lambda }_i)u_j,\tau {\dot{u}}_i=-2{\lambda }_i{u}_i$)

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 $\tau d\Vert w{\Vert }^2/dt=2(w^{T}Cw)(1-\alpha \Vert w{\Vert }^2)$.