[[LU Decomposition Remastered]] --- ### **Intro** The LDL decomposition is a general decomposition for symmetric matrices. > $A = LDL^T$ > Where $L$ is lower triangular with ones on the diagonal and $D$ is a diagonal matrix. When the symmetric matrix is positive definite, this decomposition is applicable. **Feasibility of such a thing** **Claim 1** > LDL decomposition is possible, when the matrix A is positive definite, or semi-definite. This will mean that $A = MM^T$ Take note that: $ \begin{aligned} A &= M^TM \\ A &= (QR)^T(QR) \\ A &= (R^TQ)(QR) \\ A &= R^TR \end{aligned} $ It's possible to find a $RD$ where $D$ is a diagonal matrix for every given strict upper triangular matrix $R$. More specifically: $D_{i, i} = R_{i, i}$, then in that case, we have $A = R^TD^2R$, this is the form anticipated by the $LDL$ decomposition. **Claim 2** > $LDL$ decomposition is just LU decomposition performed at 2 sides of the matrix.