This question was somewhat answered here: Fast trace of inverse of a square matrix. However, I feel like there was no complete answer wrt the Cholesky case.

I have the matrix $\Sigma=LL^T$. Is there a way of getting $Tr(\Sigma^{-1})$ without using the SVD? I'm guessing eigen decomposition is just as costly as SVD. I have already computer the lower Cholesky matrix $L$ for a previous computation.

The matrix is symmetric, positive definite and (unfortunately) dense.