Schur complement

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