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A number of theoretical and computational problems for matrix polynomials aresolved by passing to linearizations. Therefore a perturbation theory resultsfor linearizations need to be related back to matrix polynomials. In this paperwe present an algorithm that finds which perturbation of matrix coefficients ofa matrix polynomial corresponds to a given perturbation of the entirelinearization pencil. Moreover we find transformation matrices that, via strictequivalence, transform a perturbation of the linearization to the linearizationof a perturbed polynomial. For simplicity, we present the results for the firstcompanion linearization but they can be generalized to a broader class oflinearizations.

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