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, 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 A^k 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 ${\displaystyle A=(a_{ij})}$ be an ${\displaystyle n\times n}$ positive matrix: ${\displaystyle a_{ij}>0}$ for ${\displaystyle 1\leq i,j\leq n}$. Then the following statements hold.

All of these properties extend beyond strictly positive matrices to primitive matrices (see below). Facts 1–7 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 ${\displaystyle w^{T}v=1}$.

There is an extension to matrices with non-negative entries. Since any non-negative matrix can be obtained as a limit of positive matrices, 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, for the example ${\displaystyle A=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)}$, the maximum eigenvalue r = 1 has the same absolute value as the other eigenvalue −1; while for ${\displaystyle A=\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right)}$, the maximum eigenvalue is r = 0, which is not a simple root of the characteristic polynomial, and the corresponding eigenvector (1, 0) is not strictly positive.

However, Frobenius found a special subclass of non-negative matrices — irreducible matrices — for which a non-trivial generalization is possible. For such a matrix, although the eigenvalues attaining the maximal absolute value might not be unique, their structure is under control: they have the form ${\displaystyle \omega r}$, where ${\displaystyle r}$ is a real strictly positive eigenvalue, and ${\displaystyle \omega }$ ranges over the complex h' th roots of 1 for some positive integer h called the period of the matrix. The eigenvector corresponding to ${\displaystyle r}$ has strictly positive components (in contrast with the general case of non-negative matrices, where components are only non-negative). Also all such eigenvalues are simple roots of the characteristic polynomial. Further properties are described below.

Let A be a n × n square matrix over field F. The matrix A is irreducible if any of the following equivalent properties holds.