About Perron-Frobenius
Theorem (1907–1912): If A is a positive matrix (all entries > 0), then:
• There exists a unique largest real eigenvalue λ₁ > 0 (the Perron root)
• Its eigenvector has all positive components
• All other eigenvalues satisfy |λᵢ| < λ₁
Power iteration: repeatedly multiply a random vector by A and normalize. It converges geometrically to the Perron eigenvector.
Applications: Google PageRank, population dynamics, Markov chain stationary distributions, network centrality.