On Schur and pseudo-Schur complements

Abstract

The primary goal of this instructive paper is to present various useful identities and explore basic properties of the two concepts: Schur and Pseudo-Schur complements (named in honor of the mathematician Isaai Schur) which have been paid attention by some researchers in many fields of mathematics. Schur and Pseudo-Schur complements play a central role and they serve as a rich and powerful tool by many authors. I will go over numerous main formulas related with these two concepts and some of their applications (I will omit the proofs). This paper is intended to familiarize the reader with the \textit{Schur} and Pseudo-Schur complements, and this by using a lucid style that will attract readers from diverse backgrounds. The literature on the subject is vast, and its applications far reaching. It is highly recommended that one examine more rigorously the references of this paper.