  In this paper, we analyze the convergence behavior of the randomized extended
Kaczmarz (REK) method for all types of linear systems (consistent or
inconsistent, overdetermined or underdetermined, full-rank or rank-deficient).
The analysis shows that the larger the singular value of $A$ is, the faster the
error decays in the corresponding right singular vector space, and as
$k\rightarrow\infty$, $x_{k}-x_{\star}$ tends to the right singular vector
corresponding to the smallest singular value of $A$, where $x_{k}$ is the $k$th
approximation of the REK method and $x_{\star}$ is the minimum $\ell_2 $-norm
least squares solution. These results explain the phenomenon found in the
extensive numerical experiments appearing in the literature that the REK method
seems to converge faster in the beginning. A simple numerical example is
provided to confirm the above findings.
