A reduced form for perturbed matrix polynomials

ABSTRACT

We show that every perturbation A(λ, ε) of an n x n matrix polynomial A(λ) such that det A(λ) = λ^m with m ≤ n can be reduced by equivalence transforms to a perturbed matrix polynomial whose leading matrix has maximal Smith form. This yields a reduced form for square perturbed matrix polynomials from which one can easily recover all the eigenvalue leading terms of the form με^β with β^-1 ∈ ℕ^*.

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.5 Roots of Nonlinear Equations
          Subjects: Polynomials, methods for

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computation of transforms (e.g., fast Fourier transform)

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Eigenvalues and eigenvectors (direct and iterative methods)

General Terms:
Documentation, Standardization

Keywords:
Newton diagram, Smith normal form, matrix polynomials, perturbed eigenvalues