Partitioned positive definite matrix property

Question

I am interested in the following problem:

Let $X$ be a real symmetric positive definite matrix partitioned into four submatrices as follows: $$ X = \begin{pmatrix} A&B\\ B^T & C \end{pmatrix} $$ Define the symmetric matrix $Y$ by: $$ Y = C - B^TA^{-1}B $$ Is $Y$ always positive definite?

I suspect that the answer is yes. I have attempted using the definition of positive definite matrices to prove this and also tried looking for counterexamples, but have so far been unsuccessful. I would greatly appreciate any help or recommendations.

Answer 1

Yes.

Recognizing that $Y$ is the Schur complement of $A$ in $X$, we can use the following condition for positive definiteness of a Schur complement:

Let $X$ be a symmetric matrix of real numbers given by $$ X=\begin{pmatrix}A&B\\B^T&C\end{pmatrix} $$ Then if A is invertible, then X is positive definite if and only if A and its complement X/A are both positive definite: $$ {\displaystyle X\succ 0\Leftrightarrow A\succ 0,X/A=C-B^{\mathsf {T}}A^{-1}B\succ 0} $$

Since $X$ is positive definite, $A$ is positive definite and is therefore invertible. Then by the above equivalence statement, we know that $Y=C-B^TA^{-1}B$ is positive definite.