Perron-Frobenius theorem

Let A be a nonnegative matrix. Denote its spectrum by σ(A). Then the spectral radius ρ(A) is an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath, that is, ρ(A)σ(A), and is associated to a nonnegative eigenvectorMathworldPlanetmathPlanetmathPlanetmath.

If, in addition, A is an irreducible matrixMathworldPlanetmath, then |ρ(A)||λ|, for all λσ(A), λρ(A), and ρ(A) is a simple eigenvalue associated to a positive eigenvector.

If, in addition, A is a primitive matrix, then ρ(A)>|λ| for all λσ(A), λρ(A).

Title Perron-Frobenius theoremMathworldPlanetmath
Canonical name PerronFrobeniusTheorem
Date of creation 2013-03-22 13:18:26
Last modified on 2013-03-22 13:18:26
Owner jarino (552)
Last modified by jarino (552)
Numerical id 5
Author jarino (552)
Entry type Theorem
Classification msc 15A18
Related topic FundamentalTheoremOfDemography