LU decomposition

Definition 1

Let A be an n×m matrix. An LU decompositionMathworldPlanetmath (sometimes also called an LU factorization) of A, if it exists, is an n×n unit lower triangular matrix L and an n×m matrix U, in (upper) echelon formMathworldPlanetmath, such that


The LU factorization is closely related to the row reduction algorithmMathworldPlanetmath. In a very real sense, the factorization is a record of the steps taken in row reducing a matrix to echelon form. The matrix L “encodes” the sequence of row replacement operationsMathworldPlanetmath that row reduce the given matrix A to echelon form U.

Proposition 2

Suppose that A=LU is an LU factorization, and let Lij denote the entries of L. Then, the row reduction AρU is accomplished by the following sequence ρ of 12n(n-1) row replacement operations:


Note: the first n-1 steps clear out column 1, the next n-2 steps clear out column 2, etc.

Proposition 3

Not every matrix admits an LU factorization. Indeed, an LU factorization exists if and only if A can be reduced to echelon without using row exchange operations. However, if an LU factorization exists, then it is unique.

In the most general case, one has to employ row exchange operations.

Proposition 4

Let A be an n×m matrix. Then, there exists an n×n permutation matrixMathworldPlanetmath P (indeed, many such) such that the matrix PA admits an LU factorization, i.e., there exist matrices P,L,U such that


The key idea behind LU factorization is that one does not need to employ row scalings to do row reduction until the second half (the back-substitution phase) of the algorithm. This has significant implicationMathworldPlanetmath for numerical stability of the algorithm.

The LU decomposition of a given matrix A is useful for the solution of the systems of linear equations of the form Ax=b. Indeed, it suffices to first solve the linear system Ly=b, and second, to solve the system Ux=y. This two-step procedure is easy to implement, because owing to the lower and upper-triangular nature of the matrices L and U, the required row operations are determined, more or less, directly from the entries of L and U. Indeed, the first step,


a sequence of row operations ρ1, and the second step


a sequence of row operations ρ2, are exactly the same row operations one has to perform to row reduce A directly to reduced echelon form E:


Note: x0 is the particular solution of Ax=b that sits in the rightmost colulmn of the augmented matrix at the termination of the row-reduction algorithm.

Title LU decomposition
Canonical name LUDecomposition
Date of creation 2013-03-22 12:06:37
Last modified on 2013-03-22 12:06:37
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Definition
Classification msc 65-00
Classification msc 15-00
Synonym LU factorization
Synonym LU-factorization
Synonym LU-decomposition
Related topic QRDecomposition