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Perron–Frobenius theorem

In linear algebra, 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 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, Hawkins–Simon condition);to demography (Leslie population age distribution model); to social networks (DeGroot learning process), to Internet search engines and even to ranking of footballteams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. In linear algebra, 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 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, Hawkins–Simon condition);to demography (Leslie population age distribution model); to social networks (DeGroot learning process), to Internet search engines and even to ranking of footballteams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. Let positive and non-negative respectively describe matrices with exclusively positive real numbers as elements and matrices with exclusively non-negative real numbers as elements. The eigenvalues of a real square matrix A are complex numbers that make up the spectrum of the matrix. The exponential growth rate of the matrix powers Ak as k → ∞ is controlled by the eigenvalue of A with the largest absolute value (modulus). The Perron–Frobenius theorem describes the properties of the leading eigenvalue and of the corresponding eigenvectors when A is a non-negative real square matrix. Early results were due to Oskar Perron (1907) and concerned positive matrices. Later, Georg Frobenius (1912) found their extension to certain classes of non-negative matrices. Let A = ( a i j ) {displaystyle A=(a_{ij})} be an n × n {displaystyle n imes n} positive matrix: a i j > 0 {displaystyle a_{ij}>0} for 1 ≤ i , j ≤ n {displaystyle 1leq i,jleq n} . Then the following statements hold. These claims can be found in Meyer chapter 8 claims 8.2.11–15 page 667 and exercises 8.2.5,7,9 pages 668–669. The left and right eigenvectors w and v are sometimes normalized so that the sum of their components is equal to 1; in this case, they are sometimes called stochastic eigenvectors. Often they are normalized so that the right eigenvector v sums to one, while w T v = 1 {displaystyle w^{T}v=1} . An extension of the theorem to matrices with non-negative entries is also available. In order to highlight the similarities and differences between the two cases the following points are to be noted: every non-negative matrix can be obtained as a limit of positive matrices, thus one obtains the existence of an eigenvector with non-negative components; the corresponding eigenvalue will be non-negative and greater than or equal, in absolute value, to all other eigenvalues. However, the simple examples

[ "Matrix (mathematics)", "Eigenvalues and eigenvectors" ]
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