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Matrix Factorization
Undergraduate
Definition
Matrix factorization expresses a matrix as product of simpler matrices. Used for solving systems, eigenvalue computation, and data compression.
Formulas
A = LU
LU decomposition (lower × upper triangular)
A = QR
QR decomposition (orthogonal × upper triangular)
A = UΣ Vᵀ
Singular Value Decomposition (SVD)
A = LDLᵀ
Cholesky decomposition (symmetric positive definite)
Examples
Example 1
Explain how to solve Ax = b using LU decomposition.
Applications
Numerical Linear Algebra
Solving linear systems
Machine Learning
Dimensionality reduction, recommender systems
Image Processing
Compression, denoising
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