Let $X=[X_1, X_2]$ be a full column rank matrix, where $X_1$ is a $n\times p$ matrix and $X_2$ is a $n\times q$ matrix. Both $X_1$ and $X_2$ are full column rank matrices. That is to say, ${\rm rank}(X)=p+q$, ${\rm rank}(X_1)=p$ and ${\rm rank}(X_2)=q$. Prove that the matrix $X(X^TX)^{-1}X^T-X_1(X_1^TX_1)^{-1}X_1^T$ is a positive semidefinite matrix.
I try to use Woodbury inverse formula, but it seems fails. Thanks for your help:)