# LU factorization

(4.3 hours to learn)

## Summary

The LU factorization is a factorization of a matrix into a lower triangular and an upper triangular matrix. It can be computed by recording the row operations used in Gaussian elimination. It can be a more efficient and numerically stable method of solving linear systems compared to matrix inverses.

## Context

This concept has the prerequisites:

- Gaussian elimination (Gaussian elimination is used to compute the LU factorization.)
- matrix multiplication (The matrix is factorized into a product of two matrices.)
- matrix inverse (Matrix inverses are needed in deriving the factorization.)
- matrix transpose (Matrix transposes are needed in deriving the factorization.)

## Core resources (read/watch one of the following)

## -Free-

→ MIT Open Courseware: Linear Algebra (2011)

Videos for an introductory linear algebra course focusing on numerical methods.

Location:
Lecture "Factorization into A = LU"

→ A First Course in Linear Algebra (2012)

## -Paid-

→ Introduction to Linear Algebra

An introductory linear algebra textbook with an emphasis on numerical methods.

Location:
Sections 2.6, "Elimination = factorization: A = LU", and 2.7, "Transposes and permutations," pages 95-113

## See also

- If the matrix is symmetric positive definite, we never need to pivot; in this case, the factorization is known as the Cholesky decomposition .