We can take a covariance matrix $\Sigma$ and decompose this into a lower and upper triangular matrix $\Sigma = U^T U$ where $U$ is the Cholesky matrix. This matrix can be used to transform uncorrelated standard normal variables $X$ and $Y$ to $W$ and $Z$ such that the covariance of $W$ and $Z$ is $\Sigma$. 1, 2, etc.
I have found a lot of literature supporting the above, however occasionally people use the precision matrix $\Sigma^{-1}$ instead of the covariance matrix $\Sigma$ in the decomposition and I don't understand whether i) the interpretation of the resulting variables $W$ and $Z$ is the same and ii) in what situations one would use the precision matrix instead of the covariance matrix.
Edit in response to @whuber: In the situation that I have encountered (Weeks et al 2020, Polygenic Priority Scores), the decomposition of $\Sigma^{-1}$ is being used to enforce a dependency between two variables (to account for known correlations between them due to linkage disequilibrium between genes) such that the covariance of $X$ and $Y$ are $MVN(0, \Sigma)$. This is my interpretation and the understanding that I have formed so far. To be more specific, the procedure is described as modelling $Y = X\beta + \epsilon, \epsilon \sim MVN(0, \Sigma)$. If interested, the implementation is here. At a high level, this seems to be enforcing the known correlation structure, which lines up with what I have found from reading about using Cholesky decompositions (which was the motivation given above) but those references give the source being decomposed as $\Sigma$. Intuition here would suggest that the authors are doing the same thing but are instead decomposing $\Sigma^{-1}$ and I'm not sure what the effects of this may be.
Taking the decomposition of $\Sigma^{-1} = L^T L$, the procedure that the authors have used to enforce this dependency is simply to take the two standard normal variables $X$ and $Y$ and multiply them by the lower triangular matrix $L$ such that $W = LX$, $Z = LY$ and covariance of $Z$ and $W$ is now (I think) $\Sigma$. This last part is what I am not sure about though, as I would think that performing this from the decomposition of $\Sigma^{-1}$ would lead to the two variables having covariance $\Sigma^{-1}$, not $\Sigma$.
