An affine eigenvalue problem on the nonnegative orthant

In this paper, we consider the conditional affine eigenvalue problem lambda x = Ax + b, lambda is an element of R, x >= 0, parallel to x parallel to = 1, where A is an n x n nonnegative matrix, b a nonnegative vector, and parallel to center dot parallel to a monotone vector norm. Under suitable hypotheses, we prove the existence and uniqueness of the solution (lambda(*), x(*)) and give its expression as the Perron root and vector of a matrix A + bc(*)(T) where c(*) has a maximizing property depending on the considered norm. The equation x (Ax + b)/ parallel to Ax + b parallel to has then a unique nonnegative solution, given by the unique Perron vector of A + bc(*)(T). (c) 2005 Elsevier Inc. All rights reserved.