Cross Gramian

In control theory, the cross Gramian (formula_1, also referred to by formula_2) 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 formula_7 is controllable and observable, and hence minimal, if and only if the matrix formula_1 is nonsingular, (i.e. formula_1 has full rank, for any formula_10).
If the associated system formula_7 is furthermore symmetric, such that there exists a transformation formula_12 with
then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:
Thus the direct truncation of the Eigendecomposition of the cross Gramian allows model order reduction (see ) without a balancing procedure as opposed to balanced truncation.
The cross Gramian has also applications in decentralized control, sensitivity analysis, and the inverse scattering transform.