Chapter 5: Square Matrix and LU Decomposition. In the chapter mind map this icon labels LDU Decomp ($A = LDU$). The discussion below is excerpted and lightly edited from § Theorem: LDU Decomposition in Mathematics for AI and Machine Learning.

For any matrix $A \in \mathbb{R}^{N \times N}$ that admits LU decomposition, there exists a decomposition

where $L \in \mathbb{R}^{N \times N}$ is a lower triangular matrix with unit diagonal, $D \in \mathbb{R}^{N \times N}$ is a diagonal matrix, and $U \in \mathbb{R}^{N \times N}$ is an upper triangular matrix with unit diagonal.

This decomposition is obtained from LU decomposition by factoring $U = DU'$ where $D$ contains the diagonal elements of $U$ and $U'$ is the unit upper triangular matrix obtained by dividing each row of $U$ by its diagonal element.