Nonnegative iterations with asymptotically constant coefficients

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