Recovering a perturbation of a matrix polynomial from a perturbation of its first companion linearization

Abstract A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization equivalent corresponding polynomial, is needed. In this paper we develop an algorithm finds which coefficients polynomial corresponds given entire pencil. Moreover find transformation matrices that, via strict equivalence, transform perturbed polynomial. For simplicity, present results first companion but they can be generalized broader class