  The Kaczmarz method is an iterative projection scheme for solving con-sistent
system $Ax = b$. It is later extended to the inconsistent and ill-posed linear
problems. But the classical Kaczmarz method is sensitive to the correlation of
the adjacent equations. In order to reduce the impact of correlation on the
convergence rate, the randomized Kaczmarz method and randomized block Kaczmarz
method are proposed, respectively. In the current literature, the error
estimate results of these methods are established based on the error
$\|x_k-x_*\|_2$, where $x_*$ is the solution of linear system $Ax=b$. In this
paper, we extend the present error estimates of the Kaczmarz and randomized
Kaczmarz methods on the basis of the convergence theorem of Kunio Tanabe, and
obtain some general results about the error $\|x_k-P_{N(A)}x_0-x^\dagger\|_2$.
