  Randomized Kaczmarz is a simple iterative method for finding solutions of
linear systems $Ax = b$. We point out that the arising sequence
$(x_k)_{k=1}^{\infty}$ tends to converge to the solution $x$ in an interesting
way: generically, as $k \rightarrow \infty$, $x_k - x$ tends to the singular
vector of $A$ corresponding to the smallest singular value. This has
interesting consequences: in particular, the error analysis of Strohmer \&
Vershynin is optimal. It also quantifies the `pre-convergence' phenomenon where
the method initially seems to converge faster. This fact also allows for a fast
computation of vectors $x$ for which the Rayleigh quotient $\|Ax\|/\|x\|$ is
small: solve $Ax = 0$ via Randomized Kaczmarz.
