This  question was somewhat answered here: http://mathoverflow.net/questions/46553/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 $Tr(\Sigma)$ without using the SVD? I'm guessing eigen decomposition is just as costly as SVD.

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