In control theory, the cross Gramian is a Gramian matrix used to determine how controllable and observable a linear system is. For the stable time-invariant linear system the cross Gramian is defined as: and thus also given by the solution to the Sylvester equation: This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric. The triple is controllable and observable, and hence minimal, if and only if the matrix is nonsingular, (i.e. has full rank, for any ).