Schur Complements and Applications

Abstract

Schur complements arise naturally in the process of inverting block matrices of the form

$$M=\left (\begin{array}{cc} A&B\\ C &D\end{array} \right )\!$$

and in characterizing when symmetric versions of these matrices are positive definite or positive semidefinite. These characterizations come up in various quadratic optimization problems; see Boyd and Vandenberghe [1], especially Appendix B. In the most general case, pseudo-inverses are also needed.