On a fundamental property of the cross- Gramian matrix

The cross-Gramian matrix
W_{c0} (T)
for linear single-input single-output (SISO) dynamical systems has been related to the controllability and observability Gramians via
W^{2}_{c0}(T)= W_{c}(T)W_{0}(T)
for
T= \infty
. The result is now proved for any arbitrary time interval
T
, and is extended to symmetric multivariable systems which have symmetric transfer function matrices.