Suppose I have a matrix $A \in \mathbb{R}^{m\times n}$ with $m \geq n$ and suppose that a matrix $G=(A^T A)^{-1}$ exists.
Now suppose that I have an other matrix $B \in \mathbb{R}^{m\times m}$ that is a diagonal matrix and whose entries on the diagonal can only be 0 or 1
Define $C=BA$ and $H=(C^T C)^{-1}$
Is there some particular relationship between $G$ and $H$?
Thanks to everyone
This is not a complete answer, but here are some ideas.
So matrix multiplication is easily remembered as row times column. So when we calculate $A^TA$, we are actually calculating column times column. So you can think of the matrix $A^TA$ as containing all the inner products between the columns of $A$.
Now, the matrix $B$ has easy properties: $B^2=B$ and $B^T=B$. Hence $C^TC=A^TB^TBA=A^TBA$. Lets calculate the $i,j$-th element of $C^TC$. We find that $$(A^TBA)_{ij} = \sum_{k=1}^m A_{ki}B_{kk}A_{kj}.$$ Hence the $i,j$-th entry of $C^TC$ contains the inner product between the $i$-th column of $A$ and the $j$-th column of $A$, but we delete the $k$-th components of the columns when $B_{kk}=0$.
In fact this calculation shows that the matrix $C^TC$ is independent of the rows of $A$ corresponding to the (diagonal) zeroes in $B$. Thus the matrix $H$ is independent of the rows of $A$ corresponding to zeroes in $B$. But clearly, the matrix $G$ does depend on this. To me, this seems to suggest that there is no nice relation between $G$ and $H$.