--- title: LU Factorization description: LU factorization decomposes a matrix into a product of a lower triangular matrix and an upper triangular matrix. Learners will understand how to apply this decomposition to solve linear systems efficiently. category: mathematics subcategory: numerical-linear-algebra difficulty: beginner, intermediate, advanced url: /subject/lu-factorization --- # LU Factorization LU factorization decomposes a matrix into a product of a lower triangular matrix and an upper triangular matrix. Learners will understand how to apply this decomposition to solve linear systems efficiently. ## Available Resources 1 Videos • 3 Courses • 1 Websites • 1 Papers ## Courses ### 1. Linear Algebra (MIT 18.06SC) **Author:** Gilbert Strang Matrix theory and linear algebra following Strang's textbook: elimination, vector spaces, orthogonality and least squares, determinants, eigenvalues, positive definite matrices and the singular value decomposition. Includes 74 lecture and problem-solving videos, summary notes, and problem sets and exams with solutions, built for independent study. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/ **Tags:** linear-algebra, vector-spaces, eigenvalues, least-squares, singular-value-decomposition ### 2. ALAFF Chapter 5: The LU and Cholesky Factorizations **Author:** Robert van de Geijn, Margaret Myers The strongest free structured treatment that goes past the definition into the algorithmic variants an implementer actually needs. Chapter of UT Austin's graduate numerical linear algebra text that moves from Gaussian elimination to LU, then to row pivoting, complete pivoting, blocked algorithms and Cholesky, using FLAME notation with worked derivations and programming exercises. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.cs.utexas.edu/~flame/laff/alaff/chapter05-LU-factorization.html **Tags:** lu-factorization, cholesky-factorization, pivoting, blocked-algorithms, numerical-linear-algebra ### 3. Linear Algebra - Foundations to Frontiers Learn the mathematics behind linear algebra and link it to matrix software development. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.edx.org/learn/linear-algebra/the-university-of-texas-at-austin-linear-algebra-foundations-to-frontiers **Tags:** courses, mathematics-statistics, linear-algebra ## Papers ### 1. Gaussian Elimination (MIMS EPrint 2011.16) **Author:** Nicholas J. Higham Ten-page survey explaining why Gaussian elimination computes an LU factorization and what that viewpoint buys, then covering pivoting strategies, growth factors, backward error bounds, matrix classes needing no pivoting, and blocked LAPACK-style implementations. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://eprints.maths.manchester.ac.uk/1579/1/covered/MIMS_ep2011_16.pdf **Tags:** gaussian-elimination, lu-factorization, pivoting, backward-error-analysis, numerical-linear-algebra ## Videos ### 1. Lecture 4: Factorization into A = LU (MIT 18.06) **Author:** Gilbert Strang Derives A = LU from elimination, explains why the multipliers appear unchanged in L, counts elimination's cost, and introduces permutation matrices for PA = LU. One lecture video from a 35-video course with solved problem sets and exams; afterwards you can factor matrices by hand. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/lecture-4-factorization-into-a-lu/ **Tags:** lu-decomposition, gaussian-elimination, permutation-matrices, matrix-factorization, linear-algebra ## Websites ### 1. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/lu-factorization