Suppose that $A$ is a $m\times n$ full row rank sparse matrix, and $Q$ is an $n\times n$ symmetric positive definite sparse matrix with $m<n$. Besides, $m$ is about $10^5$, and $n$ is about $10^6$. There is no other special structure for $A$ and $Q$ (i.e., not circulant, not Toeplitz, etc.). I need to solve the linear system $AQ^{-1}A'X=B$, where $B$ is an $m\times m$ matrix. Is there any way to find the explicit formula for $(AQ^{-1}A')^{-1}$?
BTW, calculation of cholesky decomposition for $Q$ is not a problem on my laptop, but computing $Q^{-1}A'=L'\backslash(L\backslash A')$ breaks down with out of memory errors in MATLAB. If I compute $Q^{-1}A'$ with parfor loop in MATLAB, it is extremely slow. Direct solvers for these linear systems are preferred.
Thanks, Pulong