On inhomogeneous eigenvalue problems. I

AbstractIn studying stability of solutions of linear differential equations one naturally encounters Liapunov equations. In a suitable setting they can be interpreted as equations for the normalized “directions” of these solutions. When applying discretizations to the Liapunov equations one is led to a problem which in its most elementary form can be stated as: Given a matrix A and a vector b, determine a vector x with xTx = 1 and a scalar μ such that Ax − b = μx. Here μ is called the inhomogeneous eigenvalue. We consider the question of how many solution pairs (μ, x) of this problem exist. We also give some numerical methods to compute such a pair; they are based on (generalizations of) shifted power iterations. Finally we consider the case that ‖b‖ is small so that the inhomogeneous eigenvalues can be viewed as perturbations of the homogeneous ones (i.e. b = 0)