In control theory, the cross Gramian is a Gramian matrix used to determine how controllable and observable a linear system is.
For the stable timeinvariant linear system
x
˙
=
A
x
+
B
u
y
=
C
x
the cross Gramian is defined as:
W
X
:=
∫
0
∞
e
A
t
B
C
e
A
t
d
t
and thus also given by the solution to the Sylvester equation:
A
W
X
+
W
X
A
=
−
B
C
The triple
(
A
,
B
,
C
)
is controllable and observable if and only if the matrix
W
X
is nonsingular, (i.e.
W
X
has full rank, for any
t
>
0
).
If the associated system
(
A
,
B
,
C
)
is furthermore symmetric, such that there exists a transformation
J
with
A
J
=
J
A
T
B
=
J
C
T
then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:

λ
(
W
X
)

=
λ
(
W
C
W
O
)
.
Thus the direct truncation of the singular value decomposition of the cross Gramian allows model order reduction (see [1]) without a balancing procedure as opposed to balanced truncation.