On some structural properties of generalized Lyapunov eigenproblems and application to operator preconditioning

We are interested in generalized matrix eigenvalue problems of the type AX + XA = lambda HXH and AX + XA = lambda(HX + XH) with A and H both symmetric and positive definite, and in their tensor counterparts. We collect several structural properties, some of which are known, together with some new spectral results. We also analyze in detail the case where the second problem stems from the discretization of linear elliptic partial differential equations by finite differences. In particular, we derive spectral properties that can be used in the numerical solution of the resulting algebraic linear system.