 Solution

For fixed weights during the input average,Thus $C$ is the input second-moment matrix. It is a positive semidefinite matrix because $v^{T}Cv=⟨(v^{T}x{)}^2⟩\ge 0$. If the inputs have been mean-centered, it is their covariance matrix; otherwise $C=\mathrm{Cov}(x)+⟨x⟩⟨x{⟩}^T$.

The stationary averaged learning equation isFor $w\ne 0$ this makes $w$ an eigenvector of $C$ with eigenvalue $\lambda =\alpha ⟨y^2⟩$. If $\lambda >0$, taking the inner product with $w$ givesand thereforeThe zero vector is also stationary but is not an eigenvector in the usual nonzero-vector definition. Likewise any vector in the nullspace of $C$ is stationary with zero output second moment and need not have the normalized length. These exceptional equilibria must be distinguished from a nonzero feature-learning solution.