In matrix theory, the Perron–Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by 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, Hawkins–Simon condition); to demography (Leslie population age distribution model); to social networks (DeGroot learning process); to Internet search engines (PageRank); and even to the ranking of American football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau.
Statement 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.
Positive matrices Let A = ( a i j ) {\displaystyle A=(a_{ij})} be an n × n {\displaystyle n\times n} positive matrix: a i j > 0 {\displaystyle a_{ij}>0} for 1 ≤ i , j ≤ n {\displaystyle 1\leq i,j\leq n} . Then the following statements hold.
There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r , |λ| < r. Thus, the spectral radius ρ ( A ) {\displaystyle \rho (A)} is equal to r. If the matrix coefficients are algebraic, this implies that the eigenvalue is a Perron number. The Perron–Frobenius eigenvalue is simple: r is a simple root of the characteristic polynomial of A. Consequently, the eigenspace associated to r is one-dimensional. (The same is true for the left eigenspace, i.e., the eigenspace for AT, the transpose of A.) There exists an eigenvector v = (v1,...,vn)T of A with eigenvalue r such that all components of v are positive: A v = r v, vi > 0 for 1 ≤ i ≤ n. (Respectively, there exists a positive left eigenvector w : wT A = wT r, wi > 0.) It is known in the literature under many variations as the Perron vector, Perron eigenvector, Perron-Frobenius eigenvector, leading eigenvector, principal eigenvector or dominant eigenvector. There are no other positive (moreover non-negative) eigenvectors except positive multiples of v (respectively, left eigenvectors except w), i.e., all other eigenvectors must have at least one negative or non-real component.
… excerpt ends here. Continue reading the full article.
