Problem：
Let $X$ and $R$ be positive definite matrices, $C$ is a matrix of compatible dimension, and define $g(X)$ as $g(X)=X-XC'[CXC'+R]^{-1}CX$
Prove that if $X>Y>0$, then $g(X)>g(Y)$.
From the definition, I have the followings:
$g(X)+XC'[CXC'+R]^{-1}CX=X$
$g(Y)+YC'[CYC'+R]^{-1}CY=Y$
Moreover, $X>Y>0$ implies that
$g(X)+XC'[CXC'+R]^{-1}CX>g(Y)+YC'[CYC'+R]^{-1}CY$ and $X>0,Y>0$.
I try to use the inversion lemma, but the equation becomes more and more complicated.
I don't know how to prove that $g(X)>g(Y)$ based on the above equations.
Could you please help me?
Note that \begin{align*} X > Y > 0 &\implies 0 < X^{-1} < Y^{-1} \\ &\implies X^{-1} + C^\intercal R^{-1} C < Y^{-1} + C^\intercal R^{-1} C \\ &\implies (X^{-1} + C^\intercal R^{-1} C)^{-1} > (Y^{-1} + C^\intercal R^{-1} C)^{-1} \\ &\implies g(X) > g(Y) \end{align*} by Woodbury's formula.