(4.3 hours to learn)
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.
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)
→ 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)
A linear algebra textbook with proofs.
Location: Section "Triangular decomposition"
→ 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
- If the matrix is symmetric positive definite, we never need to pivot; in this case, the factorization is known as the Cholesky decomposition .
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