The Schur complement of a matrix is a way to "break down" into smaller blocks, which is especially useful for finding determinants or positive definiteness. For a matrix partitioned into

the Schur complement of in is

and the Schur complement of in is

The intuition behind this definition is that when solving

we can isolate the effect of by substituting, which gives us

(and similarly for isolating ).

Determinant#

Using the Schur complement, we can find the determinant of as

Positive Definiteness#

If is positive definite, and the Schur complement of in is positive definite, then symmetric (where ) is also positive definite. The reverse is also true.

Quadratic Form#

The quadratic form with symmetric matrix can be expressed in block diagonal form,