Chapter 2.4, Problem 15E

Suppose A₁₁ is invertible. Find X and Y such that

$\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]=\left[\begin{array}{cc}I& 0\\ X& I\end{array}\right]\left[\begin{array}{cc}{A}_{11}& 0\\ 0& S\end{array}\right]\left[\begin{array}{cc}I& Y\\ 0& I\end{array}\right]$    (7)

where S = A₂₂ − A₂₁ $A_{11}^{-1}$ A₁₂. The matrix S is called the Schur complement of A₁₁. Likewise, if A₂₂ is invertible, the matrix A₁₁ − A₁₂ $A_{22}^{-1}$ A₂₁ is called the Schur complement of A₂₂. Such expressions occur frequently in the theory of systems engineering, and elsewhere.