Power Iteration Method

Linear Algebra for Machine Learning · Axiom Academy

Finding the dominant eigenvalue numerically Power iteration is a simple numerical method to find the largest eigenvalue and its eigenvector. It's especially useful when you have a large matrix. Start with a random non-zero vector x₀ Repeatedly multiply by A and normalize: x_ k+1 = Ax_k / ||Ax_k|| The vector converges to the dominant eigenvector The Rayleigh quotient λ ≈ x^T Ax / (x^T x) converges to the dominant eigenvalue Find the dominant eigenvalue of: The algorithm converges to the dominant (largest magnitude) eigenvalue. The convergence rate depends on the ratio λ₁/λ₂ (larger ratio = faster convergence). Large sparse matrices: Only need to compute Ax, not invert anything Distributed computing: Can be parallelized easily Web crawling: PageRank uses power iteration to find the dominant eigenvector Streaming data: Can update iteratively as new data arrives Only finds the dominant (largest) eigenvalue Convergence can be slow if λ₁ ≈ λ₂ Requires normalized matrix operations for stability For finding multiple eigenvalues, use deflation or other methods

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