Perron-Frobenius Perron-Frobenius Theorem: A:a nonnegative nxn matrix with spectral radius (A) (A)>0 is an eigenvalue of A; there is a nonnegative (left and right)eigenvector associated with o(A); if further A is irreducible,then: there is a positive (left and right)eigenvector associated with o(A)that is of multiplicity 1; for stochastic matrix A the spectral radius o(A)=1.Perron-Frobenius • A : a nonnegative n×n matrix with spectral radius ρ(A) • ρ(A) > 0 is an eigenvalue of A; • there is a nonnegative (left and right) eigenvector associated with ρ(A); • if further A is irreducible, then: • there is a positive (left and right) eigenvector associated with ρ(A) that is of multiplicity 1; • for stochastic matrix A the spectral radius ρ(A)=1. Perron-Frobenius Theorem: