# Perron-Frobenius theorem

Let $A$ be a nonnegative matrix. Denote its spectrum by $\sigma(A)$. Then the spectral radius $\rho(A)$ is an eigenvalue, that is, $\rho(A)\in\sigma(A)$, and is associated to a nonnegative eigenvector.

If, in addition, $A$ is an irreducible matrix, then $|\rho(A)|\geq|\lambda|$, for all $\lambda\in\sigma(A)$, $\lambda\neq\rho(A)$, and $\rho(A)$ is a simple eigenvalue associated to a positive eigenvector.

If, in addition, $A$ is a primitive matrix, then $\rho(A)>|\lambda|$ for all $\lambda\in\sigma(A)$, $\lambda\neq\rho(A)$.

Title Perron-Frobenius theorem PerronFrobeniusTheorem 2013-03-22 13:18:26 2013-03-22 13:18:26 jarino (552) jarino (552) 5 jarino (552) Theorem msc 15A18 FundamentalTheoremOfDemography