On the Meany inequality with applications to convergence analysis of several row-action iteration methods

The Meany inequality gives an upper bound in the Euclidean norm for a product of rank-one projection matrices. In this paper we further derive a lower bound related to this inequality. We discuss the internal relationship between the upper bounds given by the Meany inequality and by the inequality in Smith et al. (Bull Am Math Soc 83:1227-1270, 1977) in the finite dimensional real linear space. We also generalize the Meany inequality to the block case. In addition, by making use of the block Meany inequality, we improve existing results and establish new convergence theorems for row-action iteration schemes such as the block Kaczmarz and the Householder-Bauer methods used to solve large linear systems and least-squares problems.