An affine eigenvalue problem on the nonnegative orthant

Blondel, Vincent;Ninove, Laure;Van Dooren, Paul
(2005) Linear Algebra and Its Applications — Vol. 404, p. 69-84 (2005)

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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.
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Blondel, V., Ninove, L., & Van Dooren, P. (2005). An affine eigenvalue problem on the nonnegative orthant. Linear Algebra and Its Applications, 404, 69-84. https://doi.org/10.1016/j.laa.2005.02.036 (Original work published 2005)