Schur complement

Given a block matrix M = [A, B; C, D] with A invertible, the Schur complement of A is S = D - C A^-1 B. It is what is left of the block D after you eliminate the block C using A as the pivot. You can read it as the effective behavior of the second block once the influence of the first has been folded in.

It is the single most useful quantity in block linear algebra, because three different questions all reduce to it. Invertibility: M is invertible exactly when both A and S are invertible. Determinant: det(M) = det(A) det(S). And the (2,2) block of M^-1 is precisely S^-1, so S governs how the second variable responds after the first is solved out.

Definiteness inherits too. A symmetric matrix M is positive definite if and only if A is positive definite and its Schur complement S is positive definite. This is the engine behind testing positive definiteness blockwise and behind the Cholesky and LDL^T factorizations, which are just repeated Schur-complement steps.

It carries real meaning across fields. In statistics, the Schur complement of a covariance block is the conditional covariance of one set of Gaussian variables given the other. In optimization it appears in KKT and saddle-point systems. And because S can be far worse conditioned than M, it is also a warning sign about numerical sensitivity hidden inside a benign-looking matrix.