Perturbation of eigenvalues of matrix pencils and optimal assignment problem

We consider a matrix pencil whose coefficients depend on a positive parameter $ϵ$, and have asymptotic equivalents of the form $a{ϵ}^A$ when $ϵ$ goes to zero, where the leading coefficient $a$ is complex, and the leading exponent $A$ is real. We show that the asymptotic equivalent of every eigenvalue of the pencil can be determined generically from the asymptotic equivalents of the coefficients of the pencil. The generic leading exponents of the eigenvalues are the “eigenvalues” of a min-plus matrix pencil. The leading coefficients of the eigenvalues are the eigenvalues of auxiliary matrix pencils, constructed from certain optimal assignment problems.