Given the matrix $X_{n,p}$ from the least squares problem $$ \mathbf{X} \cdot \mathbf{\beta} = z $$
Where the normal equation is:
$$ \mathbf{\hat{\beta}} = \left(\mathbf{X}^T \mathbf{X}\right)^{-1} \mathbf{X}^T z $$
I was very happy when I found the existence of the Woodbury matrix identity unfortunantly I am struggling to use it (don't know if it's possible) for my problem.
$$ {(A+UCV)}^{-1}=A^{-1}-A^{-1}U{(C^{-1}+VA^{-1}U)}^{-1}VA^{-1} $$
The Problem
I want to compute a new $(X^TX)^-1$ after removing the first $k$ rows of $X$. I heard maybe it's called the leave-one-out (k-out?) statistics.
I found that for the my case the Woodbury formula is something like:
$$ {((X^TX)+UCV)}^{-1}=(X^TX)^{-1}-(X^TX)^{-1}U{(C^{-1}+V(X^TX)^{-1}U)}^{-1}V(X^TX)^{-1} $$
where $+UCV$ should somehow subtract the first $k$ rows.
If someone can give some help or point to some direction or references.