The Perron-Frobenius theorem: every irreducible nonneg matrix A has a unique largest real eigenvalue λ₁ > 0 (Perron root) with strictly positive eigenvector v₁ > 0. All other eigenvalues satisfy |λᵢ| ≤ λ₁. The power method demonstrates this: x_{k+1} = Ax_k / ‖Ax_k‖ converges to v₁ regardless of initial vector. For row-stochastic matrices (Markov chains), λ₁ = 1. Applications: PageRank (Google), Markov chain steady states, population dynamics (Leslie matrices), and community detection.