Hermitian quadratic matrix polynomials: Solvents and inverse problems

A monic quadratic Hermitian matrix polynomial L(λ) can be factorized into a product of two linear matrix polynomials, say L(λ)=(Iλ-S)(Iλ-A). For the inverse problem of finding a quadratic matrix polynomial with prescribed spectral data (eigenvalues and eigenvectors) it is natural to prescribe a right solvent A and then determine compatible left solvents S. This problem is explored in the present paper. The splitting of the spectrum between real eigenvalues and nonreal conjugate pairs plays an important role. Special attention is paid to the case of real-symmetric quadratic polynomials and the allocation of the canonical sign characteristics as well as the eigenvalues themselves.