  Our aim in this paper is two-fold: First, for computing zeros of a linear
time-invariant (LTI) system $\Sigma$ in {\em state-space form}, we introduce a
"trimmed structured linearization", which we refer to as {\em Rosenbrock
linearization}, of the Rosenbrock system polynomial $\mathcal{S}(\lam)$
associated with $\Sigma.$ We also introduce Fiedler-like matrices for
$\mathcal{S}(\lam)$ and describe constructions of Fiedler-like pencils for
$\mathcal{S}(\lam).$ We show that the Fiedler-like pencils of
$\mathcal{S}(\lam)$ are Rosenbrock linearizations of the system polynomial
$\mathcal{S}(\lam).$ Second, with a view to developing a direct method for
solving rational eigenproblems, we introduce "linearization" of a rational
matrix function. We describe a state-space framework for converting a rational
matrix function $G(\lam)$ to an "equivalent" matrix pencil $\mathbb{L}(\lam)$
of smallest dimension such that $G(\lam)$ and $\mathbb{L}(\lam)$ have the same
"eigenstructure" and we refer to such a pencil $\mathbb{L}(\lam)$ as a
"linearization" of $G(\lam).$ Indeed, by treating $G(\lam)$ as the transfer
function of an LTI system $\Sigma_G$ in state-space form via state-space
realization, we show that the Fiedler-like pencils of the Rosenbrock system
polynomial associated with $\Sigma_G$ are "linearizations" of $G(\lam)$ when
the system $\Sigma_G$ is both controllable and observable.
