I have a formula $Qy = v^TA^{-1}v$ with vector $v = y - \xi$ for a fixed $\xi$, and covariance matrix $A$ (see the formula for $Qy$ at the bottom of page 21 in appendix A.2). I know that for reasons of speed and accuracy, it's generally good to avoid inverting matrices (see here for example). Is there any way to avoid that when computing $Qy$ here?
I currently find the inverse of $A$ by using $AX=I$, where $I$ is the identity matrix, and a solver and solving for $X$. I'm hoping there's some way to rearrange things that avoids the need for even this method of inversion. In particular, I'm interested in methods likely to perform better when implemented in numpy/scipy (the $AX=I$ approach uses cho_solve and isn't particularly fast.).
Unless $V$ has some special relationship to $A$ (as noted by Arthur) I don't think there's anything better than solving $AX = V$ and then setting $Q = V^TX$.
This is slightly faster than your current approach (since you only need to solve one linear system instead of $n$).