Author(s) :
Leonhard Batzke
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 362014
MSC 2000
 15A22 Matrix pencils

47A55 Perturbation theory
Abstract :
The spectral behavior of regular Hermitian matrix pencils is examined under certain
structurepreserving rank1 and rank2 perturbations.
Since Hermitian pencils have signs attached to real (and infinite) blocks in canonical form,
it is not only the Jordan structure but also this socalled sign characteristic
that needs to be examined under perturbation.
The observed effects are as follows: Under a rank1 or rank2
perturbation, generically the largest one or two, respectively, Jordan blocks at each
eigenvalue $\lambda$ are destroyed, and if $\lambda$ is an eigenvalue of the perturbation, also
one new block of size one is created at $\lambda$.
If $\lambda$ is real (or infinite), additionally all signs at $\lambda$ but one or two, respectively,
that correspond to the destroyed blocks, are preserved under perturbation.
Also, if the potential new block of size one is real, its sign is in most cases prescribed
to be the sign that is attached to the eigenvalue $\lambda$ in the perturbation.
Keywords :
Matrix pencil, Hermitian matrix pencil, sign characteristic, rank one perturbation, rank two perturbation, generic perturbation