Perron–Frobenius theorem

In linear algebra, the Perron–Frobenius theorem, proved by OskarPerron (1907) and GeorgFrobenius (1912), asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Leontief's input-output model); to demography (Leslie population age distribution model), to Internet search engines and even ranking of football teams. The first to discuss the ordering of players within tournaments using Perron-Frobenius eigenvector is Edmund Landau.