Schurkomplement

The Schur complement is a concept in linear algebra, named after the German mathematician Issai Schur. It is used to express the inverse of a partitioned matrix in terms of its submatrices. This technique is particularly useful in various fields such as control theory, statistics, and numerical analysis.

Consider a square matrix A partitioned as follows:

A = [ B C

D E ]

where B, C, D, and E are submatrices. The Schur complement of E in A, denoted as S_E(A), is defined as:

S_E(A) = B - C * E^(-1) * D

Here, * denotes matrix multiplication, and E^(-1) is the inverse of E. The Schur complement has several important properties. If A is invertible, then the Schur complement of E in A is also invertible, and its inverse is given by:

S_E(A)^(-1) = E^(-1) + E^(-1) * C * (B - C * E^(-1) * D)^(-1) * C * E^(-1)

The Schur complement can be used to simplify the computation of the inverse of a partitioned matrix. It is also useful in the context of block matrices and in the analysis of matrix pencils.