Block LU Factorization
From the book
Chapter 10: Matrix Block Partitioning. In the chapter mind map this icon labels Block Decomps: Block LU/ Cholesky/QR. The discussion below is excerpted and lightly edited from § Theorem: Block Cholesky Decomposition in Mathematics for AI and Machine Learning.
Let $A \in \mathbb{R}^{N \times N}$ be symmetric positive definite, partitioned as
where $A_{00} \in \mathbb{R}^{m \times m}$ is positive definite. Then $A$ can be decomposed as
where $L_{00} L_{00}^\top = A_{00}$ (Cholesky of $A_{00}$), $L_{10} = A_{10} (L_{00}^\top)^{-1}$, and $L_{11} L_{11}^\top = A_{11} - L_{10} L_{10}^\top$ (Cholesky of the Schur complement).
What this drawing shows
What you see. Depicts a matrix partitioned into blocks for LU-style elimination, useful for explaining block Gaussian elimination and Schur complements.
In the mind map. Chapter 10 — Block Decomps: Block LU/ Cholesky/QR. 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 10: Matrix Block Partitioning — see the mind-map node Block Decomps: Block LU/ Cholesky/QR.