This paper presents a comprehensive picture of the mapping of structural properties associated with general linear multivariable
systems under bilinear transformation. While the mapping of poles of
linear multivariable systems under such a transformation is well
known, the question of how the structural invariant properties of a
given system are mapped remains unanswered. This paper addresses this
question. More specifically, we investigate how the finite and
infinite zero structures, as well as invertibility structures, of a
general continuous-time (discrete-time) linear time-invariant
multivariable system are mapped to those of its discrete-time
(continuous-time) counterpart under the bilinear (inverse bilinear)
transformation. We demonstrate that the structural invariant indices
lists ${cal I}_2$ and ${cal I}_3$ of Morse remain invariant under
the bilinear transformation, while the structural invariant indices
lists ${cal I}_1$ and ${cal I}_4$ of Morse are, in general, changed.