The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. To social networks ( DeGroot learning process) to Internet search engines ( PageRank) and even to ranking of football To demography ( Leslie population age distribution model) 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, Hawkins–Simon condition ) The corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. 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.
