Generic Asymptotics of Eigenvalues Using Min-plus Algebra

Abstract: We consider a square matrix Aǫ whose entries have first order asymptotics of the form (Aǫ)i j ∼ ai j ǫAi j when ǫ goes to 0, for some ai j ∈ C and Ai j ∈ R. We show that under a non-degeneracy condition, the order of magnitudes of the different eigenvalues ofAǫ are given by min-plus eigenvalues of min-plus Schur complements built from A = (Ai j ), or equivalently by generalized minimal mean weights of circuits. This construction gives, in non singular cases, a graph interpretation to the slopes of the Newton polygon of the characteristic polynomial of Aǫ . It explains the order of magnitudes of eigenvalues in the perturbation formula of Lidskiı̆, Višik and Ljusternik, and it allows us to solve some cases which are singular in this theory. Copyright 2001 IFAC