Inhalt des Dokuments
Preprint 31-2006
Structured eigenvalue condition number and backward error of a class of polynomial eigenvalue problems
Author(s) :
Shreemayee Bora
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 31-2006
MSC 2000
- 65F15 Eigenvalues, eigenvectors
-
65F35 Matrix norms, conditioning, scaling
Abstract :
We consider the normwise
condition number and backward error of eigenvalues of matrix
polynomials having $\star$-palindromic/antipalindromic and
$\star$-even/odd structure with respect to structure preserving
perturbations. Here $\star$ denotes either the transpose $T$ or
the conjugate transpose $*.$ We show that when the polynomials are
complex and $\star$ 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.
Keywords :
Nonlinear eigenvalue problem,palindromic matrix polynomial, odd matrix polynomial, even matrix polynomial, structured eigenvalue condition number,structured backward error