DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-30

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Block LU factorisation and the associated Schur complement

Definition

Let

$A = (\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array})$

be a square matrix over a field, with square leading block $A_{11}$ .

A block LU factorisation across this partition is a factorisation

$A = (\begin{array}{cc} L_{11} & 0 \\ L_{21} & L_{22} \end{array}) (\begin{array}{cc} U_{11} & U_{12} \\ 0 & U_{22} \end{array}),$

where the diagonal blocks are square, the left factor is block lower triangular, and the right factor is block upper triangular.

If $A_{11}$ is invertible, the Schur complement of $A_{11}$ in $A$ is

$S : = A_{22} - A_{21} A_{11}^{- 1} A_{12} .$

This is the trailing block left after eliminating the off-diagonal blocks using the invertible leading block.

Depends on

Used by

An invertible leading block yields block LU through its Schur complement Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David Bindel, CS 4220: Numerical Analysis, Blocked LU and Cholesky (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Section 2.6 (standard reference, not scraped)