Schur complement

In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. Suppose A, B, C, D are respectively p × p, p × q, q × p, and q × q matrices, and D is invertible. Let so that M is a (p + q) × (p + q) matrix. Then the Schur complement of the block D of the matrix M is the p × p matrix defined by and, if A is invertible, the Schur complement of the block A of the matrix M is the q × q matrix defined by In the case that A or D is singular, substituting a generalized inverse for the inverses on M/A and M/D yields the generalized Schur complement. The Schur complement is named after Issai Schur who used it to prove Schur's lemma, although it had been used previously. Emilie Haynsworth was the first to call it the Schur complement. The Schur complement is a key tool in the fields of numerical analysis, statistics and matrix analysis. The Schur complement arises as the result of performing a block Gaussian elimination by multiplying the matrix M from the right with a block lower triangular matrix Here Ip denotes a p×p identity matrix. After multiplication with the matrix L the Schur complement appears in the upper p×p block. The product matrix is