Before revision: For the stable time-invariant linear system the cross Gramian is defined as: and thus also given by the solution to the Sylvester equation:
After revision: This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric.For the stable time-invariant linear system the cross Gramian is defined as: and thus also given by the solution to the Sylvester equation: