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 (= G0 and invertible). The spectrum of A0 is real when A0 is G0-strongly d efinitizable. 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 neutr ality'' 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 perturbations of (suitably defined) definitizable matrix polynomi als. Applications are made to quadratic matrix polynomials arising in the study of damped systems and gyroscopic systems.