# Find first element of inverse matrix knowing Cholesky decomposition.

Given Cholesky decomposition of matrix A = LDL$$^{T}$$ = $$A^{T}$$ provide a possibly most efficient method to calculate upper left element of $$A^{-1}$$. I was thinking that this could be solved by using Gaussian algorithms, but couldn't really get anything out of it.

You want to compute $$e_1^TL^{-T}D^{-1}L^{-1}e_1=(L^{-1}e_1)^TD^{-1}(L^{-1}e_1)$$ which means you have to solve the lower left triangular system $$Lv=e_1$$ and then compute $$v^TD^{-1}v=\sum d_k^{-1}v_k^2$$.