On inhomogeneous eigenvalue problems. I.

In 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).