Nonnegative iterations with asymptotically constant coefficients

AbstractLet Ak,k=0,1,2,…, be a sequence of real nonsingular n×n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties Au=λu and ‖u‖=1. Under further additional conditions on the spectrum of A, it is shown that if x0≠0 and the iteratesxk+1=Akxk,k=0,1,2,…,are nonnegative, then xk‖xk‖ converges to u and ‖xk+1‖‖xk‖ converges to λ as k→∞