 Solution

Averaging the fast fluctuations in $yx=x{x}^{T}w$ givesThus a nonzero steady vector satisfiesIts direction is an eigenvector of the input second-moment matrix, and its magnitude is the square root of that eigenvalue. The zero vector is also a steady state and satisfies the displayed relation with $\lambda =0$. This is an averaged steady state; the microscopic fluctuating update need not vanish at each instant.