  Zeros of rational transfer function matrices $R(\lambda)$ are the eigenvalues
of associated polynomial system matrices $P(\lambda)$, under minimality
conditions. In this paper we define a structured condition number for a simple
eigenvalue $\lambda_0$ of a (locally) minimal polynomial system matrix
$P(\lambda)$, which in turn is a simple zero $\lambda_0$ of its transfer
function matrix $R(\lambda)$. Since any rational matrix can be written as the
transfer function of a polynomial system matrix, our analysis yield a
structured perturbation theory for simple zeros of rational matrices
$R(\lambda)$. To capture all the zeros of $R(\lambda)$, regardless of whether
they are poles or not, we consider the notion of root vectors. As corollaries
of the main results, we pay particular attention to the special case of
$\lambda_0$ being not a pole of $R(\lambda)$ since in this case the results get
simpler and can be useful in practice. We also compare our structured condition
number with Tisseur's unstructured condition number for eigenvalues of matrix
polynomials, and show that the latter can be unboundedly larger. Finally, we
corroborate our analysis by numerical experiments.
