Definition:LU Factorization

Definition

Let $\mathbf A$ be a square matrix.

An LU factorization of $\mathbf A$ is a factorization of $\mathbf A$ in the form:

$\mathbf A = \mathbf {L U}$

where:

$\mathbf L$ is a lower triangular matrix and $\mathbf U$ is a upper triangular matrix.

It is usual to arrange this so that either $\mathbf L$ or $\mathbf U$ has a unit diagonal.

Examples

Arbitrary Example

The matrix:

$\begin {pmatrix} 2 & 1 \\ -1 & 2 \end {pmatrix}$

has this LU factorization:

$\begin {pmatrix} 1 & 0 \\ -1 & 1 \end {pmatrix} \begin {pmatrix} 2 & 1 \\ 0 & 3 \end {pmatrix}$

Also see

• Results about LU factorizations can be found here.

Linguistic Note

An LU factorization is so called from the initial letters of the lower triangular matrix and upper triangular matrix that compose it.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factorization: 2. (of a matrix)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): LU factorization
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factorization: 2. (of a matrix)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): LU factorization