# Schur complement

From Encyclopedia of Mathematics

2020 Mathematics Subject Classification: *Primary:* 15A24 [MSN][ZBL]

Let $M$ be a square matrix over a fixed ground field, partitioned in block form as $$ M = \left({ \begin{array}{cc} P & Q \\ R & S \end{array} }\right) \ , $$ where $P$ is a square non-singular submatrix.

The complement of $P$ is $$ M/P = S - R P^{-1} Q \ . $$

The Schur determinant lemma may be expressed in the form $$ \det(M) = \det(P) \det(M/P) \ . $$

#### References

- Hogben, Leslie.
*Handbook of linear algebra*(2nd enlarged ed.) Discrete Mathematics and its Applications, Chapman & Hall/CRC (2014)**ISBN**978-1-4665-0728-9 Zbl 1284.15001 - Zhang, Fuzhen (ed.)
*The Schur complement and its applications*, Numerical Methods and Algorithms**4**Springer (2005)**ISBN**0-387-24271-6Zbl 1075.15002

**How to Cite This Entry:**

Schur complement.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Schur_complement&oldid=54384