Перрон-Фробениус теоремасы және оның қолданыстары

Theory in linear algebra
In matrix theory, the Perron–Frobenius theorem, proved by and , 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, Hawkins–Simon condition);
to demography (Leslie population age distribution model);
to social networks (DeGroot learning process); to Internet search engines (PageRank); and even to ranking of American football
teams. 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 and concerned positive matrices. Later, found their extension to certain classes of non negative matrices.