Perron–Frobenius theorem

The Perron–Frobenius theorem states that a matrix with strictly positive entries has a positive eigenvalue equal to its spectral radius and a corresponding strictly positive eigenvector.

Spectral radius: The largest absolute value of a matrix’s complex eigenvalues. The theorem identifies a positive eigenvalue whose value is exactly this maximum.
Eigenvalue: A scalar by which a linear transformation scales some nonzero vector. The theorem singles out one eigenvalue with maximal absolute value.
Google PageRank: A link-analysis algorithm that ranks web pages using the stationary distribution of a modified link matrix. Its ranking vector is a positive dominant eigenvector of the Google matrix.
Perron–Frobenius theorem for nonnegative matrices: An extension of Perron–Frobenius theory to matrices with nonnegative entries, allowing reducibility and periodicity. It relaxes strict positivity but requires care about uniqueness and other spectral conclusions.
Oskar Perron: A German mathematician who proved foundational results about positive matrices in 1907. His work established the positive-matrix result now bearing his name.
Positive matrix: A real matrix whose entries are all strictly greater than zero. Strict positivity is the hypothesis that yields the theorem’s strongest conclusions.
Eigenvector: A nonzero vector whose direction is unchanged by a linear transformation. Its special eigenvector also has every coordinate strictly positive.
Leslie matrix: A matrix model of age-structured population growth using survival and fertility rates. Its dominant eigenvalue gives long-term population growth under standard positivity conditions.
Primitive matrix: A nonnegative matrix some positive integer power of which has all entries strictly positive. The theorem’s strongest spectral conclusions extend to primitive matrices.
Georg Frobenius: A German mathematician who extended Perron’s results to nonnegative matrices. His extensions made irreducibility and periodicity central to the theorem’s broader theory.

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Spectral graph theoryRelated: It underpins interpretations of leading adjacency eigenvectors in connected graphs.