• Overview
• Details
• Exercises

Matrix factorizations express a matrix as a product of simpler matrices. LU and QR aid solving linear systems and least squares. The SVD reveals intrinsic structure and is foundational for PCA and low-rank approximation.

• LU: A = PLU with permutation P to ensure stability; forward/back substitution to solve Ax=b efficiently for many b.
• QR: A = QR with Q orthogonal and R upper-triangular; Householder vs Gram–Schmidt; least squares via R.
• SVD: A = UΣV^T with singular values σ_i ≥ 0; rank r equals number of positive σ_i; best rank-k approximation by truncating Σ.
• Conditioning: relation of σ_min/σ_max to condition number κ; numerical stability concerns.
• Applications: PCA, dimensionality reduction, pseudoinverse A^+ = VΣ^+U^T, low-rank compression.

1. Compute the LU decomposition of a 3×3 matrix (with row pivots) and solve Ax=b for two right-hand sides.
2. Use Householder reflections to form the QR of a tall matrix and solve a least-squares problem.
3. Compute the truncated SVD (k=2) of a small matrix and compare reconstruction error vs full A.