Linear and multilinear algebra; matrix theory Concept

LU decomposition

Summary

An LU decomposition of a square matrix $A$ is a factorization $A = LU$ . Here $L$ is a lower triangular matrix and $U$ is an upper triangular matrix. Gaussian elimination without row exchanges produces such a factorization: $U$ is the row echelon form, and $L$ stores the multipliers of the row operations. With row exchanges, one gets $PA = LU$ for a permutation matrix $P$ , and each square matrix over a field has such a factorization.

Remarks

An invertible matrix has an LU decomposition with $L$ unit lower triangular if and only if all its leading principal minors are nonzero. This decomposition is then unique. The matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has no LU decomposition, but it has one after a row exchange. To solve $Ax = b$ , solve $Ly = b$ by forward substitution and $Ux = y$ by back substitution. The decomposition also gives $\det A = \det L \det U$ .

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Tag	`0RKG5YW66Y`
URI	`m2://0RKG5YW66Y`
Aliases	`lu-factorization`
Kind	concept
Writing	W1 stated
MSC	15A23, 65F05
Same as	Wikidata Q833089 Wikipedia LU decomposition

LU decomposition

Summary

Remarks

Where this leads

Further reading

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