On the spectra of sums of orthogonal projections with applications to parallel computing

Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the convergence of a related sequential method is determined by the spectrum of the product of complementary projections. The paper studies spectral properties of sums of orthogonal projections and in the case of two projections, characterizes the spectrum of the sum completely in terms of the spectrum of the product.