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) ⁻¹ R,A=P+L (Lower)+R (upper), then Seidel iteration ofAX=Y ₀ 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 ] ²₁ , whereA ₂₁=−A ^*₁₂ ,A ₁₁,A ₂₂ are invertible;A ₁₁ +A ^*₁₁ =A₂₂+A ^*₂₂ ; and the proper values ofA ⁻¹₁₂ A ₁₁,A ^*−1₁₂ A ₂₂ are in the interior of the unit disk.