Variational principle: R(x) = xᵀAx / xᵀx is minimized/maximized by eigenvectors
Rayleigh Quotient: For a symmetric matrix A, R(x) = xᵀAx / xᵀx satisfies λ_min ≤ R(x) ≤ λ_max for all non-zero x. Equality holds exactly when x is an eigenvector. This gives a variational characterization of eigenvalues (Courant-Fischer min-max theorem). Power iteration repeatedly applies A and renormalizes: x ← Ax/|Ax|, converging to the dominant eigenvector at rate |λ₂/λ₁|. Inverse iteration applies (A−σI)⁻¹ to converge to the eigenvalue nearest σ. The Rayleigh quotient iteration updates σ = R(xₖ) each step, giving cubic convergence.