Power method: starting from v₀, iterate vₖ₊₁ = Avₖ/||Avₖ||. The Rayleigh quotient λₖ = vₖᵀAvₖ/vₖᵀvₖ converges to the dominant eigenvalue. Rate: |λ₂/λ₁|ᵏ. Used in Google's PageRank.
How to Use
Set up the matrix:
- A: Square matrix
- v₀: Initial vector
- Output: λ₁ + eigenvector
PageRank Connection
Google PageRank = dominant eigenvector of the link matrix. Power method on billions of web pages. Convergence rate depends on ratio |λ₂/λ₁|, which Google controls via damping factor.
Limitations
Only finds dominant eigenvalue. Fails if |λ₁| = |λ₂|. Slow if λ₂ ≈ λ₁. For all eigenvalues, use QR algorithm. For specific eigenvalues, use inverse iteration or shift-invert.
Step-by-Step Instructions
- 1Select a matrix.
- 2View iterations.
- 3Watch λ converge.
- 4See eigenvector.
- 5Check convergence rate.