On the numerical solution of AX - XB = C

In this note we analyze the numerical solution of AX - XB = C when a Galerkin method is applied, assuming that B has much smaller size than A. We show that the corresponding Galerkin equation can be obtained from the truncation of the original problem, if matrix polynomials are used for writing the analytical solution X. Moreover, we provide some relations between the separation of A and B in their natural space and that in the projected space. Experimental tests validate some of the theoretical results and show the rate of applicability of the method with respect to a standard linear system solver.