gen-define-the-singular-value-decomposition-and-give-one-short-w-98aa

Define the singular value decomposition and give one short worked example.

# Singular Value Decomposition (SVD) ## Definition For any matrix **A** ∈ ℝ^(m×n), the **Singular Value Decomposition** is a factorization: **A = UΣV^T** where: - **U** ∈ ℝ^(m×m) is orthogonal (U^T U = I) with columns called left singular vectors - **Σ** ∈ ℝ^(m×n) is diagonal with non-negative entries σ₁ ≥ σ₂ ≥ ... ≥ 0 called singular values - **V** ∈ ℝ^(n×n) is orthogonal (V^T V = I) with columns called right singular vectors ## Worked Example Find the SVD of **A** = [3 0] [0 2] **Step 1:** Compute **A^T A** A^T A = [3 0] [3 0] = [9 0] [0 2] [0 2] [0 4] **Step 2:** Find eigenvalues and eigenvectors of **A^T A** Eigenvalues: λ₁ = 9, λ₂ = 4 Eigenvectors: v₁ = [1], v₂ = [0] [0] [1] So **V** = [1 0] [0 1] **Step 3:** Singular values are σ₁ = √9 = 3, σ₂ = √4 = 2 **Σ** = [3 0] [0 2] **Step 4:** Compute **U** from u_i = (1/σᵢ)Avᵢ u₁ = (1/3)[3 0][1] = [1] [0 2][0] [0] u₂ = (1/2)[3 0][0] = [0] [0 2][1] [1] So **U** = [1 0] [0 1] **Result:** A = [1 0][3 0][1 0] [0 1][0 2][0 1]