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A normalised LU factorisation has a unit lower-triangular L and an upper-triangular U
Definition
Let be a field, let , and let .
A normalised LU factorisation of is a factorisation
where is lower triangular with diagonal entries all equal to , and is upper triangular. Thus is unit lower triangular.
When every diagonal entry of is nonzero, the factorisation has nonzero pivots. Later existence and uniqueness results state explicitly when that extra hypothesis is required.
Depends on
Used by
- An LDU factorisation has unit lower-triangular L, diagonal D, and unit upper-triangular U Definition
- Block LU factorisation and the associated Schur complement Definition
- A square matrix has an unpivoted unit-lower LU factorisation exactly when all leading principal minors are nonzero Theorem
- Every real or complex square matrix admits a PLU factorisation, and the first failed pivot marks the singular boundary Theorem
- Normalised LU and LDU factorisations with nonzero pivots are unique Theorem
- When unpivoted elimination runs to completion, the elimination multipliers are the entries of the normalised LU factors Theorem
Dependency tree · two levels
5 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
- Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Section 2.6 (standard reference, not scraped)
- David Bindel, CS 4220: Numerical Analysis, Blocked LU and Cholesky (standard reference, not scraped)