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.
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics