Gaussian Elimination (MIMS EPrint 2011.16)
by Nicholas J. Higham · University of Manchester (MIMS)
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.
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ALAFF Chapter 5: The LU and Cholesky Factorizations
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.
Lecture 4: Factorization into A = LU (MIT 18.06)
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.
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