Matrix Factorization & Decomposition

Goal: Find unit vector w that maximises Var(wᵀX) = wᵀΣw, subject to ||w||=1
Lagrangian: L = wᵀΣw − λ(wᵀw − 1). Set ∂L/∂w = 0
Result: 2Σw − 2λw = 0 → Σw = λw — this IS the eigenvalue equation!
Conclusion: PC1 = eigenvector of Σ with largest eigenvalue λ₁. The variance captured = λ₁.
PC2, PC3, ...: Repeat after removing PC1's contribution. Each subsequent PC = next eigenvector (orthogonal to previous).

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Matrix Factorization

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