Two-by-Two Block Matrices and Their Factorization

The topics that are covered in this chapter are as follows. We first study some fundamental properties of block matrices and their Schur complements. We next consider a popular product iteration method, and then the concept of approximate block-factorization is introduced. A main relation between a familiar product iteration algorithm and a basic block-factorization preconditioner is then established. This relation is a cornerstone in proving the spectral equivalence estimates. Next, a sharp spectral equivalence result is proved in a general setting. It provides necessary and sufficient conditions in an abstract form for a preconditioner to be spectrally equivalent to the given matrix. Then two major examples, a two-level and a two-grid preconditioner, are considered and analyzed. Finally the more classical two-by-two (two-level) block-factorization preconditioners are introduced and analyzed. The chapter concludes with a procedure to generate a stable block form of matrices and with an analysis of a respective block-factorization preconditioner.