SVD from Eigendecomposition

Linear Algebra for Machine Learning · Axiom Academy

Complete Worked Example: Computing SVD of a 2×2 Matrix Find the SVD of the following 2×2 matrix using the eigendecomposition method: Question 1: What is the trace of A T A? Step 2: Find Eigenvalues of A T A Question 2: What are the eigenvalues in decreasing order? Step 3: Compute Singular Values Singular values are square roots of eigenvalues: Step 4: Find Eigenvectors of A T A For λ₁ = 9, solve (A T A - 9I)v₁ = 0: This gives eigenvector v₁ = (1/√2, 1/√2)ᵀ For λ₂ = 3, similarly: v₂ = (-1/√2, 1/√2)ᵀ V matrix (eigenvectors as columns): Using u₁ = Av₁/σ₁ and u₂ = Av₂/σ₂: Question 3: Verify the SVD by computing UΣV T . Does it equal A?

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.