Perturbation Theory for HomogeneousPolynomial Eigenvalue

We consider polynomial eigenvalue problems P(A; ;)x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (;) 2 C 2. In this framework innnite eigenvalues are on the same footing as nite 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 deene the condition numbers of a simple eigenvalue (;) 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.