Low rank perturbations of strongly definitizable transformations and matrix polynomials

Let A₀ be a transformation on a finite dimensional Hilbert space which is self-adjoint in an indefinite scalar product generated by G₀ (G^*₀ and invertible). The spectrum of A₀ is real when A₀ is G₀-strongly definitizable. The problems considered here concern the number of real eigenvalues of a G-self-adjoint transformation A where A and G are low rank perturbations of A₀ and G₀. A notion called the "order of neutrality" of A with respect to G is introduced which is relevant to this problem area. Using linearization as well as direct methods, results are obtained concerning self-adjoint matrix polynomials which are low rank perturbationsof (suitably defined) definitizable matrix polynomials. Applications are made to quadratic matrix polynomials arising in the study of damped systems and gyroscopic systems.