Schur complement

The Schur complement of a block matrix, encountered in linear algebra and the theory of matrices, is defined as follows.

Suppose p, q are nonnegative integers such that p + q > 0, and suppose A, B, C, D are respectively p × p, p × q, q × p, and q × q matrices of complex numbers. Let $${\displaystyle M=\left[{\begin{matrix}A&B\\C&D\end{matrix}}\right]}$$ so that M is a (p + q) × (p + q) matrix.

If D is invertible, then the Schur complement of the block D of the matrix M is the p × p matrix defined by $${\displaystyle M/D:=A-BD^{-1}C.}$$ If A is invertible, the Schur complement of the block A of the matrix M is the q × q matrix defined by $${\displaystyle M/A:=D-CA^{-1}B.}$$ 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^[1] who used it to prove Schur's lemma, although it had been used previously.^[2] Emilie Virginia Haynsworth was the first to call it the Schur complement.^[3] The Schur complement is a key tool in the fields of numerical analysis, statistics, and matrix analysis. The Schur complement is sometimes referred to as the Feshbach map after a physicist Herman Feshbach.^[4]