Perturbation theory for homogeneous polynomial eigenvalue problems

We consider polynomial eigenvalue problems P(A,α,β)x=0 in which the matrix polynomial is homogeneous in the eigenvalue (α,β) ∈ℂ 2. In this framework infinite eigenvalues are on the same footing as finite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is well-posed when its eigenvalues are simple. We define the condition numbers of a simple eigenvalue (α,β) and a corresponding eigenvector x and show that the distance to the nearest ill-posed problem is equal to the reciprocal of the condition number of the eigenvector x. We describe a bihomogeneous Newton method for the solution of the homogeneous polynomial eigenvalue problem (homogeneous PEP). © 2002 Elsevier Science Inc.