Zeros and poles of matrix transfer functions and their dynamical interpretation

The given rational matrix transfer function H(\cdot) is viewed as a network function of a multiport. The no X ni matrix H(s) is factored into
D_{l}(S)^{-1} N_{l}(s) = N_{r}(s)D_{r}(s)^{-1}
,where
D_{l}(\cdot),N_{l}(\cdot),N_{r}(\cdot)
, and
D_{r}(\cdot)
are polynomial matrices of appropriate size, with
D_{l}(\cdot)
and
N_{i}(\cdot)
left coprime and
N_{r}(\cdot)
and
D_{r}(\cdot)
right coprime. A zero of
H(\cdot)
is defined to be a point
z
where the local rank of
N_{l}(\cdot)
drops below the normal rank. The theorems make precise the intuitive concept that a multiport blocks the transmission of signals proportional to
e^{zt}
if and only if
z
is a zero of
H(\cdot)
. We show that p is a pole of
H(\cdot)
if and only if some "singular" input creates a zero-state response of the form
re^{pt}
, for
t > 0
. The order m of the zero z is similarly characterized. Although these results have state-space interpretation, they are derived by purely algebraic techniques, independently of state-space techniques. Consequently, with appropriate modifications, these results apply to the sampled-data case.