Multiply any starting arrow by a matrix, rescale it, and repeat — it swings toward one special direction and stops: the DOMINANT eigenvector, the direction the matrix stretches most. Its growth factor is the largest eigenvalue. This one trick ranks the whole web (PageRank is power iteration on the link matrix) and finds principal components. Slide to iterate and watch the arrow lock on.
Power iteration finds the dominant eigenpair of a matrix A by repeatedly applying it and normalizing: vₙ₊₁=Avₙ/‖Avₙ‖. Any starting vector is a mix of eigenvectors; each multiplication scales each component by its eigenvalue, so the LARGEST-magnitude one grows fastest and dominates — v converges to the dominant eigenvector and the Rayleigh quotient vᵀAv to its eigenvalue λ₁. Convergence rate is |λ₂/λ₁| (slower when the top two are close). It is the engine of PageRank (the web link matrix), PCA’s first component, and spectral methods. A fail-loud self-check throws unless it recovers a known dominant eigenvalue to 1e−6. ◆ real numerical methods, node-verified.
It finds only the DOMINANT eigenpair, converges slowly when λ₁≈λ₂, and needs a nonzero component along the top eigenvector; deflation/shifts get the rest. The iteration and Rayleigh-quotient convergence are exact.