List of formulas related to Cholesky decomposition Definition The Cholesky factor of positive-definite symmetric matrix $X$ is $$ X = L L^\top $$ where $L$ is a lower triangular matrix. Trace Theorem The trace of $X$ is $$ Tr(X) = \sum_{i,j}L_{i,j}^2 $$ Proof $$ Tr(X) = \sum_{i}X_{i,i} = \sum_{i}\left(\sum_{j}L_{i,j}L_{j,i}^\top\right) = \sum_{i,j}L_{i,j}^2 $$ Determinant Theorem The determinant of $X$ is $$ \mathrm{det}(X) = \prod_{i}L_{i,i}^2 $$ Proof $$ \mathrm{det}(X) = \mathrm{det}(L) \times \mathrm{det}(L^\top) = \prod_{i}L_{i,i} \times \prod_{i}L_{i,i} = \prod_{i}L_{i,i}^2 $$