An affine eigenvalue problem on the nonnegative orthant Original Research Article

In this paper, we consider the conditional affine eigenvalue problemimagewhere A is an n × n nonnegative matrix, b a nonnegative vector, and short parallel·short parallel a monotone 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 image, where c* has a maximizing property depending on the considered norm. The equation x = (Ax + b)/short parallelAx + bshort parallel has then a unique nonnegative solution, given by the unique Perron vector of image.