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