LU Decomposition

LU Decomposition is the factorization (when it exists) of a matrix A in terms of a unit lower triangular (L) and an upper triangular (U) matrix. So if a matrix can be reduced to it's Row-Echelon-Form (REF) using only operations of the form "add to ", then is the REF of , and is the Matrix Multiplication of the Inverse Elementary Matrices used to get into REF.

Note:
LU Decomposition doesn't always work. For example: requires a row swap. In cases like this we use a method called PLU Decomposition which is not covered in MATH 307 - Applied Linear Algebra.

Example

Go to Find LU Decomposition to see a worked example of finding the matrices given an initial matrix .

Properties

since is in REF and by definition of Rank the ranks should be identical.
For a Square Matrix , the this is because and since 's determinant is due to the lower triangular nature of the matrix. This means that meaning:
The main motivation for LU Decomposition is to solve multiple linear systems with the same but different without having to perform Gaussian Elimination each time.

Given and such that we can define a new variable such that . We can then write:This turns one difficult linear equation into two simpler linear equations allowing us to solve the system easier. Furthermore, once you have found you don't need to perform gaussian elimination if you'd like to solve a separate system with the same but different .

Note

does not need to be a square matrix in order to do LU decomposition. In the case of a rectangular matrix , and will take the following form: and .

Example

Go to Solve Using LU Decomposition to see a worked example of using LU Decomposition to solve a linear system of equations.