Numerical solution of singular Sylvester equations

We consider the numerical solution of the large scale singular Sylvester equation of the form AX−XD⊤=E (or AXB⊤−CXD⊤=E), where the spectra Λ(A) and Λ(D) (or, Λ(A−λC) and Λ(D−λB)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided.