Structured Eigenvalue Condition Number and Backward Error of a Class of Polynomial Eigenvalue Problems

We consider the normwise condition number and backward error of eigenvalues of matrix polynomials having ⋆-palindromic/antipalindromic and ⋆-even/odd structure with respect to structure preserving perturbations. Here ⋆ denotes either the transpose T or the conjugate transpose ∗. We show that when the polynomials are complex and ⋆ denotes complex conjugate, then to each of the structures there correspond portions of the complex plane so that simple eigenvalues of the polynomials lying in those portions have the same normwise condition number when subjected to both arbitrary and structure preserving perturbations. Similarly approximate eigenvalues of these polynomials belonging to such portions have the same backward error with respect to both structure preserving and arbitrary perturbations. Identical results hold when ∗ is replaced by the adjoint with respect to any sesquilinear scalar product induced by a Hermitian or skew-Hermitian unitary matrix. The eigenvalue symmetry of T -palindromic or T -antipalindromic polynomials, is with respect to the numbers 1 or −1 while that of T -even or T -odd polynomials is with respect to the origin. We show that except under certain conditions when 1, −1 and 0 are always eigenvalues of these polynomials, in all other cases their structured and unstructured condition numbers as simple eigenvalues of the corresponding polynomials are equal. The structured and unstructured backward error of these numbers as approximate eigenvalues of the corresponding polynomials are also shown to be equal. These results easily extend to the case when T is replaced by the transpose with respect to any bilinear scalar product that is induced by a symmetric or skew symmetric orthogonal matrix. In all cases the proofs provide appropriate structure preserving perturbations to the polynomials.