Editing Schur complement (section)

{{Short description|Tool in linear algebra and matrix analysis}}
The '''Schur complement''' is a key tool in the fields of [[linear algebra]], the theory of [[matrix (mathematics)|matrices]], numerical analysis, and statistics.

It is defined for a [[block matrix]]. 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
<math display="block">M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}</math>
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
<math display="block">M/D := A - BD^{-1}C.</math>
If ''A'' is invertible, the Schur complement of the block ''A'' of the matrix ''M'' is the ''q'' × ''q'' matrix defined by
<math display="block">M/A := D - CA^{-1}B.</math>
In the case that ''A'' or ''D'' is [[singular matrix|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]]<ref>{{ cite journal
|author=Schur, J.
|title=Über Potenzreihen die im Inneren des Einheitskreises beschränkt sind
|journal=J. reine u. angewandte Mathematik
|volume=147
|year=1917
|pages=205–232
|doi=10.1515/crll.1917.147.205
|url=https://eudml.org/doc/149467
}}
</ref> who used it to prove [[Schur's lemma]], although it had been used previously.<ref name="Zhang 2005">{{cite book  |title=The Schur Complement and Its Applications |first=Fuzhen |last=Zhang |editor1-first=Fuzhen |editor1-last=Zhang |series=Numerical Methods and Algorithms |year=2005 |volume=4 |publisher=Springer| isbn=0-387-24271-6 |doi=10.1007/b105056}}</ref>  [[Emilie Virginia Haynsworth]] was the first to call it the ''Schur complement''.<ref>Haynsworth, E. V., "On the Schur Complement", ''Basel Mathematical Notes'', #BNB 20, 17 pages, June 1968.</ref> The Schur complement is sometimes referred to as the ''Feshbach map'' after a physicist [[Herman Feshbach]].<ref>{{ cite journal
|author=Feshbach, Herman
|title=Unified theory of nuclear reactions
|journal=Annals of Physics
|volume=5
|issue=4
|year=1958
|pages=357–390
|doi=10.1016/0003-4916(58)90007-1
}}
</ref>