Schur Complement

The Schur complement is a key tool in linear algebra, the theory of matrices, numerical analysis, and statistics, defined for a block matrix M composed of blocks A, B, C, and D of compatible dimensions. When the block D is invertible, the Schur complement of D in M is A minus B times D inverse times C, and when A is invertible instead, the Schur complement of A in M is D minus C times A inverse times B; for singular blocks, a generalized Schur complement can be formed using generalized inverses. The concept is named after the mathematician Issai Schur, who used it to prove Schur's lemma; the name itself was given by Emilie Virginia Haynsworth, and the object is sometimes called the Feshbach map after the physicist Herman Feshbach, whose work led to that alternate name. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/