Low rank perturbations of strongly definitizable transformations and matrix polynomials

Abstract

Let A 0 be a transformation on a finite dimensional Hilbert space which is self-adjoint in an indefinite scalar product generated by G 0 ( G ∗ 0 and invertible). The spectrum of A 0 is real when A 0 is G 0-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 A 0 and G 0. 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.