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Given a positive definite matrix $\Sigma$, how can I compute the Cholesky decomposition of $\Sigma^{-1}$ from the Cholesky decomposition of $\Sigma$?

I know that $\left(L L^T \right)^{-1} = L^{-T}L^{-1}$, but is there a way to directly compute the lower triangular matrix of the Cholesky decomposition of $\Sigma^{-1}$ from $L$?

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Since $L$ is lower triangular, so is $L^{-1}$ and thus $L^{-T}$ is upper triangular. If you would rather have LU instead of UL, I do not think that there is a satisfactory way that would not be just as much or more computational work than just recalculating. I could be wrong, but I don't think so.

Maybe you deal with UL instead of LU? Or deal with the similarity after commuting? $$LU \sim UL$$

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