Constrained Matrix Sylvester Equations

The problem of finding matrices L and T satisfying $TA - FT = LC$ and $TB = 0$ is considered. Existence conditions for the solution are established and an algorithm for computing the solution is derived. Conditions under which the matrix $[C^T ,T^T ]$ is full rank are also discussed. The problem arises in control theory in the design of reduced-order observers that achieve loop transfer recovery.