What is the matrix $(X'\Omega^{-1}X)$ in Generalized Least Square / Weighted Least Square? What is the matrix $(X'\Omega^{-1}X)$ X) in Generalized Least Square / Weighted Least Square?

More precisely,
We know:


As it happens, if we have an $n \times k$ matrix $X$, $X'X = \sum_{i=1}^nx_i x_i'$, where $x_i$ is the $i^{th}$ row of $X$.  How to get from there to where you want to be should be obvious.
As a historical note, in the early days of computers (the 1970s), SAS Institute built the first widely used stats software, in part based on this relationship.  $X$ matrices of any substantial size were too big to fit into memory along with everything else - it wasn't until 1971 that the IBM 360 could have 32K of memory, and 1976 when the high-end version of the IBM 370 reached 1MB, with most computers having far less - which meant calculating $X'X$ the obvious way wasn't possible.   This relationship allowed SAS to read large (for the time) matrices in small blocks of rows and sum up the outer products $x_ix_i'$ to get $X'X$.  This allowed SAS to solve regression problems that were far larger than could fit into memory!
