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 ,where , and are polynomial matrices of appropriate size, with and left coprime and and right coprime. A zero of is defined to be a point where the local rank of drops below the normal rank. The theorems make precise the intuitive concept that a multiport blocks the transmission of signals proportional to if and only if is a zero of . We show that p is a pole of if and only if some "singular" input creates a zero-state response of the form , for . 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.