Parameter-Dependent Rank-One Perturbations of Singular Hermitian Or Symmetric Pencils

Structure-preserving generic low-rank perturbations are studied for classes of structured matrix pencils, including real symmetric, complex symmetric, and complex Hermitian pencils. For singular pencils it is analyzed which characteristic quantities stay invariant in the perturbed canonical form, and it is shown that the regular part of a structured matrix pencil is not affected by generic perturbations of rank one. When the rank one perturbations involve a scaling parameter, the behavior of the canonical forms in dependence of this parameter is analyzed as well.