Hermitian quadratic eigenvalue problems of restricted rank

Abstract

we consider a quadratic eigenvalue problem such that the second order term is a Hermitian matrix of rank r, the linear term is the identity matrix, and the constant term is an arbitrary Hermitian matrix A E cnn. Of the n + T solutions that this problem admits, we show at least n - r to be real and located in specific intervals defined by the eigenvalues of A, whence at most 2r are nonreal occuring in possibly repeated conjugate pairs.