Nonnegative iterations with asymptotically constant coefficients

Abstract

Let 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 A u = λ u and {norm of matrix} u {norm of matrix} = 1. Under further additional conditions on the spectrum of A, it is shown that if x0 ≠ 0 and the iteratesxk + 1 = Ak xk, k = 0, 1, 2, ...,are nonnegative, then frac(xk, {norm of matrix} xk {norm of matrix}) converges to u and frac({norm of matrix} xk + 1 {norm of matrix}, {norm of matrix} xk {norm of matrix}) converges to λ as k → ∞. © 2009 Elsevier Inc. All rights reserved.