LDL^T Decomposition
From the book
Chapter 9: Matrix Decompositions and Beyond. In the chapter mind map this icon labels LDL Decomp: $A = LDL^\top$. The discussion below is excerpted and lightly edited from § Theorem: LDL Decomposition in Mathematics for AI and Machine Learning.
For a __symmetric__ matrix $A \in \mathbb{R}^{N \times N}$ that admits LU decomposition, there exists a decomposition
where $L_1 \in \mathbb{R}^{N \times N}$ is a lower triangular matrix with unit diagonal, and $D \in \mathbb{R}^{N \times N}$ is a diagonal matrix. We use the notation $L_1$ (with subscript 1) to emphasize that it has unit diagonal, distinguishing it from $L$ used in other decompositions (e.g., Cholesky) where the diagonal may not be 1.
This is the symmetric version of LDU decomposition, where the symmetry of $A$ implies $U = L_1^\top$.
What this drawing shows
What you see. Shows a symmetric matrix factored into lower-triangular, diagonal, and transpose factors without square roots.
In the mind map. Chapter 9 — LDL Decomp. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 9: Matrix Decompositions and Beyond — see the mind-map node LDL Decomp.