Perron–Frobenius theorem
Theory in linear algebra / From Wikipedia, the free encyclopedia
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In matrix theory, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real. 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,[1] Hawkins–Simon condition[2]); to demography (Leslie population age distribution model);[3] to social networks (DeGroot learning process); to Internet search engines (PageRank);[4] and even to ranking of American football teams.[5] The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau.[6][7]