= Solution
For fixed weights during the input average,
$$
\langle yx\rangle=\langle xx^T\rangle w=Cw,\qquad
\langle y^2\rangle=w^TCw.
$$
Thus $C$ is the input second-moment matrix. It is a <positive semidefinite matrix> because $v^TCv=\langle(v^Tx)^2\rangle\geq0$. If the inputs have been mean-centered, it is their <covariance matrix>; otherwise $C=\operatorname{Cov}(x)+\langle x\rangle\langle x\rangle^T$.
The stationary averaged learning equation is
$$
\boxed{Cw=\alpha\langle y^2\rangle w.}
$$
For $w\ne0$ this makes $w$ an <eigenvector> of $C$ with <eigenvalue> $\lambda=\alpha\langle y^2\rangle$. If $\lambda>0$, taking the inner product with $w$ gives
$$
\lambda\|w\|^2=\alpha(w^TCw)\|w\|^2,
\qquad w^TCw=\lambda\|w\|^2,
$$
and therefore
$$
\boxed{\|w\|^2=1/\alpha.}
$$
The 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.
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