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Canonical forms for matrix triples $(A,G,\hat G)$, where
$A$ is arbitrary rectangular and $G$, $\hat G$ are either real symmetric
or skew symmetric, or complex Hermitian or skew Hermitian, are derived.
These forms generalize classical product Schur forms as well as
singular value decompositions.
An new proof for the complex case is given, where there is no need to
distinguish whether $G$ and $\hat G$ are Hermitian or skew Hermitian.
This proof is independent from the results in Bolschakov/Reichstein 1995, where
a similar canonical form has been obtained for the complex case,
and it allows generalization to the real case. Here,
the three cases, i.e., that
$G$ and $\hat G$ are both symmetric, both skew symmetric or one each,
are treated separately.
Alternating matrix polynomials, that is, polynomials whose coefficients
alternate between symmetric and skew-symmetric matrices,
generalize the notions of even and odd scalar polynomials.
We investigate the Smith forms of alternating matrix polynomials,
showing that each invariant factor is an even or odd scalar polynomial.
Necessary and sufficient conditions
are derived for a given Smith form to be that of an alternating matrix polynomial.
These conditions allow a characterization of the possible Jordan structures
of alternating matrix polynomials,
and also lead to necessary and sufficient conditions
for the existence of structure-preserving strong linearizations.
Most of the results are applicable to singular as well
as regular matrix polynomials.
We study the perturbation theory of structured matrices under
structured rank one perturbations, and then focus on several classes of complex
matrices. Generic Jordan structures of perturbed matrices are identified.
It is shown that the perturbation
behavior of the Jordan structures
is substantially different from the corresponding theory for unstructured generic
rank one perturbations.
New perturbation results for the behavior of eigenvalues and Jordan forms of real and complex matrices
under generic rank one perturbations are discussed. Several results that are available in the complex
case are proved as well for the real case and the assumptions on the genericity are weakened.
Rank one perturbations that lead to maximal algebraic multiplicities of the ``new" eigenvalues are also
discussed.
We discuss Möbius transformations for general matrix polynomials over arbitrary
elds, analyzing their in
uence on regularity, rank, determinant, constructs such as compound
matrices, and on structural features including sparsity and symmetry. Results on
the preservation of spectral information contained in elementary divisors, partial multiplicity
sequences, invariant pairs, and minimal indices are presented. The eect on canonical
forms such as Smith forms and local Smith forms, on relationships of strict equivalence
and spectral equivalence, and on the property of being a linearization or quadratication
are investigated. We show that many important transformations are special instances
of Möbius transformations, and analyze a Möbius connection between alternating and
palindromic matrix polynomials. Finally, the use of Möbius transformations in solving
polynomial inverse eigenproblems is illustrated.