Optimal rates of convergence of matrices with applications
  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. 
Heinz H. Bauschke, J.Y. Bello Cruz, Tran T.A. Nghia, Hung M. Phan, and   Xianfu Wang
math.OC math.NA