Inverse problems for Hermitian quadratic matrix polynomials

We consider the factorization of Hermitian quadratic matrix polynomials with nonsingular leading coefficient, with special emphasis on the case of real symmetric systems. It is assumed that the quadratic has a nonsingular leading coefficient (rather than the more familiar positive definite hypothesis). Theorems 3 and 8 are the central results showing how to complete a quadratic matrix function when half of the spectral data is specified. Subsequent results give more detailed information on classical problems in which the leading coefficient is positive definite. They concern the distribution of the two distinctive types of real eigenvalues. The theory is illustrated with an informative set of numerical examples.