An affine eigenvalue problem on the nonnegative orthant

In this paper, we consider the conditional affine eigenvalue problem λx = Ax + b, λ ∈ R, x ≥ 0, ‖x‖ = 1, where A is an n × n nonnegative matrix, b a nonnegative vector, and ‖·‖ a vector norm. Under suitable hypotheses, we prove the existence and uniqueness of the solution (λ∗, x∗) and give its expression as the Perron root and vector of a matrix A + bc∗ , where c∗ has a maximizing property depending on the considered norm. The equation x = (Ax+ b)/‖Ax+ b‖ has then a unique nonnegative solution, given by the unique Perron vector of A + bc∗ .