Perturbations of symmetricmatrix polynomials and theirlinearization

Abstract: The canonical stucture information, i.e. the elementary divisors and minimalindices of a matrix polynomial, is sensitive to perturbations of the matrixcoefficients of the polynomial, e.g., the eigenvalues may change or disappear.Passing to a strong linearization is a way to solve a number of problems formatrix polynomials, the linearization then has the same finite and infiniteelementary divisors and the change in minimal indices is known. However,when the linearization is perturbed by a full perturbation the correspondencebetween the linearization and matrix polynomial is lost, hence weseek a method to restore a matrix polynomial that corresponds to perturbedlinearization. Therefore we present a numerical method for computing theperturbation of a matrix polynomial from a full perturbation of its linearization.Our method is iterative and requires of solving a system of coupledSylvester equations. We limit the method to symmetric matrix polynomials.