Partitioned matrices and Seidel convergence

Summary Without using spectral resolution, an elementary proof of convergence of Seidel iteration. The proof is based on the lemma (generalizing a lemma of P. Stein): If (A+A^*)–B^*(A+A^*)B>0, whereB=–(P+L)^–1R,A=P+L (Lower)+R (upper), then Seidel iteration ofAX=Y₀ converges if and only ifA+A^*>0. This lemma has as corollaries not only the well-known results of E. Reich and Stein, but also applications to a matrix that can be far from symmetric, e.g.M=[A_ij]₁², whereA₂₁=–A₁₂^*,A₁₁,A₂₂ are invertible;A₁₁+A₁₁^*=A₂₂+A₂₂^*; and the proper values ofA₁₂^–1A₁₁,A₁₂^*–1A₂₂ are in the interior of the unit disk.Supported under NSF GP 32527.Supported under NSF GP 8758.