Is $Var(X)-Cov(X,Y)[Var(Y)]^{-1}Cov(Y,X)$ positive semidefinite?

Question

I need to prove that $Var(X)-Cov(X,Y)[Var(Y)]^{-1}Cov(Y,X)$ is positive semidefinite, where $X,Y$ are random vectors that may have different dimensions. I believe that it is a generalization of Cauchy's inequlity, but I know very little about this kind of inequalities.

Answer 1

I think the easiest way to do this is to use the Schur complement formula, which states for an invertable matrix $Z$, with block decomposition $$Z=\begin{bmatrix}A & B\\ C & D \end{bmatrix},$$ then the upper-left block of $Z^{-1}$ is given by $(A-BD^{-1}C)^{-1}$.

Considering $Z$ to be the covariance matrix of the joint vector $(X,Y)$, we see then the quantity $(\text{Var}(X)-\text{Cov}(X,Y)\text{Var}(Y)^{-1}\text{Cov}(Y,X))^{-1}$ coinsides with the upper-right block of $Z^{-1}$, with respect to the block decomposition from $(X,Y)$. As $Z$ is positive definite, we see that $Z^{-1}$ is as well, and as a submatrix of a positive definite, as well as an their inverse, is also positive definite, we can conclude that $\text{Var}(X)-\text{Cov}(X,Y)\text{Var}(Y)^{-1}\text{Cov}(Y,X)$ is positive definite.