  We investigate the randomized Kaczmarz method that adaptively updates the
stepsize using readily available information for solving inconsistent linear
systems. A novel geometric interpretation is provided which shows that the
proposed method can be viewed as an orthogonal projection method in some sense.
We prove that this method converges linearly in expectation to the unique
minimum Euclidean norm least-squares solution of the linear system, and provide
a tight upper bound for the convergence of the proposed method. Numerical
experiments are also given to illustrate the theoretical results.
