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, and suppose A, B, C, D are respectively p × p, p × q, q × p, and q × q matrices of complex numbers. Let
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
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[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]
^Schur, J. (1917). "Über Potenzreihen die im Inneren des Einheitskreises beschränkt sind". J. reine u. angewandte Mathematik. 147: 205–232. doi:10.1515/crll.1917.147.205.
^Zhang, Fuzhen (2005). Zhang, Fuzhen (ed.). The Schur Complement and Its Applications. Numerical Methods and Algorithms. Vol. 4. Springer. doi:10.1007/b105056. ISBN 0-387-24271-6.
^Haynsworth, E. V., "On the Schur Complement", Basel Mathematical Notes, #BNB 20, 17 pages, June 1968.
^Feshbach, Herman (1958). "Unified theory of nuclear reactions". Annals of Physics. 5 (4): 357–390. doi:10.1016/0003-4916(58)90007-1.
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) = {\displaystyle (D-CA^{-1}B)=} a much smaller matrix called the Schurcomplement of A {\displaystyle A} . This is the FKF method that may make it computationally...
}{\begin{pmatrix}0&0\\0&Q^{\frac {*}{2}}\end{pmatrix}},} where the Schurcomplement of A {\displaystyle {\begin{matrix}A\end{matrix}}} in the block matrix...
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Global Information![{\displaystyle M=\left[{\begin{matrix}A&B\\C&D\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06975df014f40e74936e5d74d528ec791a10d21e)
