For a matrix , we call the Schur complement of in . Note that naturally appear in block matrix inversion.
Note that when is symmetric and is positive definite, is positive definite if and only if is also positive definite. The proof is rather straightforward. Consider the function
Let’s try to minimize w.r.t. ,
Set the derivative to zero and we get
. And the minimum is
. Now, we see that for to be positive definite, for all and . This holds if and only if for all , which in turn means that has to be positive definite. The argument works for positive semidefinite as well.