M4-L4: Applications of Eigenvalues
Eigenvalues power Google, compress your data, predict crowd behavior, and keep bridges from resonating. Theory meets the real world.
Google's Secret: Dominant Eigenvector
Model the web as a directed graph. Build link matrix A where each column is a probability distribution over outgoing links. The PageRank vector r is the eigenvector of M with eigenvalue 1 — the stationary distribution of a random surfer.
Power Iteration
- Start with a uniform vector: r₀ = [1/n, …, 1/n]ᵀ
- Multiply by M repeatedly: rₖ₊₁ = Mrₖ
- Normalize each step, r ← r/‖r‖, to keep a probability vector
- Converges to the dominant eigenvector (λ = 1) in ~50 iterations
- Used by Google on billions of pages at web scale
PCA: Eigenvalues of Covariance
The covariance matrix Σ is real symmetric and positive semidefinite, so it is always orthogonally diagonalizable: Σ = QDQᵀ. Each eigenvalue λᵢ measures the variance captured along its eigenvector, and the eigenvectors — orthogonal, ordered by λ — are the principal axes of the data cloud.
Compress with k Components
Keep only the top k eigenvectors. Project all data onto these k directions. Reconstruct with a low-rank approximation — most information preserved, massive storage savings.
Long-Run Behavior: Stationary Distribution
Transition matrix P models random state jumps. Starting from any distribution, the chain converges to π — the eigenvector with eigenvalue 1. Pπ = π means one more step leaves the distribution unchanged. Weather model: P = [[0.9, 0.5], [0.1, 0.5]] gives π = [5/6, 1/6] — 83% of days sunny.
- Eigenvalue 1 always exists when every column of P sums to 1
- Perron-Frobenius: an irreducible chain has a unique π; if it is aperiodic too, any start converges to it
Natural Frequencies of Structures
Free vibration leads to the generalized eigenvalue problem Kx = ω²Mx. Each eigenvalue λᵢ gives a natural frequency ωᵢ = √λᵢ; the eigenvector gives the mode shape. Each mode tells engineers how the structure deforms at that frequency.
Resonance and Engineering
- Excitation at natural frequency → catastrophic amplitude growth
- Tacoma Narrows Bridge (1940) collapsed from resonance with wind
- Eigenvalue analysis reveals dangerous natural frequencies
- Structural engineers must ensure no overlap with expected forcing
- Damping is added to suppress resonance peaks in critical designs
Finding the Dominant Eigenvector
Power iteration: just repeated matrix-vector multiply plus normalize. Each pass scales every eigen-component by its own eigenvalue, so the dominant direction gains a factor |λ₁/λ₂| on its closest rival — equivalently, what is left of the other directions shrinks by |λ₂/λ₁| every step. Scalable to billions of rows.
Module 4 Complete
You've mastered the eigenvalue story: from solving Ax = λx, to finding eigenvectors, to diagonalization, to real applications. Linear algebra is alive in the world.