Spectral Decomposition of Self-adjoint Quadratic Pencils and Its Applications

Spectral decomposition is of fundamental importance in many applications. Generally speaking, spectral decomposition provides a canonical representation of a linear operator over a vector space in terms of its eigenvalues and eigenfunctions. The canonical form often facilitates discussions which, otherwise, would be complicated and involved. This paper generalizes the classical results of eigendecomposition for self-adjoint linear pencils, L(λ) = λ − A or Bλ − A, to self-adjoint quadratic pencils Q(λ) = Mλ2 + Cλ + K. It is shown that the decomposition involves, in addition to the usual eigeninformation, certain free parameters. These parameters occur in such an intriguing way that properly selected parameters have a variety of interesting applications. AMS subject classifications. 65F15, 15A22, 65F18, 93B55