This came up in a math reading group, so I figured I’d write a note. This is relatively computational, but I will try to keep it accessible to someone who has had a first course in linear algebra.

The Schur Complement comes up a lot when talking about a block matrix

where A is an block and D is , (thus M is . Then the Schur complement, if D is invertible, is .

The reason for how often it appears, is most readily appears explained by the factorization

which simply comes from first (upside-down) row reducing and then (upside down) column reducing.

This factorization has a few immediate results. One of the most common is that . Compare with the formula for matrices, by setting m=n=1. So M is invertible if and only if D and are both invertible.

This also has the benefit of giving a factorization of

So the Schur complement comes up pretty much whenever you want to invert a block matrix.

Another less intuitive place where this comes up is in quadratic forms. For this case, assume that M is symmetric, and hence A and D are symmetric and . If we want to write a quadratic form , in terms of a block vector

for a fixed . Then we have the following formula

To prove this formula for symmetric M, inverting the decomposition above

and

Thus

And so