A link between the Perron–Frobenius theorem and Perron’s theorem for difference equations

Let An,n∈NAn,n∈N, be a sequence of k×kk×k matrices which converge to a matrix A   as n→∞n→∞. It is shown that if xn,n∈Nxn,n∈N, is a sequence of nonnegative nonzero vectors such thatxn+1=Anxn,n∈N, then ρ=limn→∞‖xn‖n is an eigenvalue of the limiting matrix A with a nonnegative eigenvector. This result implies the weak form of the Perron–Frobenius theorem and for the class of nonnegative solutions it improves the conclusion of a Perron type theorem for difference equations.