  We present a systematic study on the linear convergence rates of the powers
of (real or complex) matrices. We derive a characterization when the optimal
convergence rate is attained. This characterization is given in terms of
semi-simpleness of all eigenvalues having the second-largest modulus after 1.
We also provide applications of our general results to analyze the optimal
convergence rates for several relaxed alternating projection methods and the
generalized Douglas-Rachford splitting methods for finding the projection on
the intersection of two subspaces. Numerical experiments confirm our
convergence analysis.
