  In this paper we derive new sufficient conditions for a linear system matrix
$$S(\lambda):=\left[\begin{array}{ccc} T(\lambda) & -U(\lambda) \\ V(\lambda) &
W(\lambda) \end{array}\right],$$ where $T(\lambda)$ is assumed regular, to be
strongly irreducible. In particular, we introduce the notion of strong
minimality, and the corresponding conditions are shown to be sufficient for a
polynomial system matrix to be strongly minimal. A strongly irreducible or
minimal system matrix has the same structural elements as the rational matrix
$R(\lambda)= W(\lambda) + V(\lambda)T(\lambda)^{-1}U(\lambda)$, which is also
known as the transfer function connected to the system matrix $S(\lambda)$. The
pole structure, zero structure and null space structure of $R(\lambda)$ can be
then computed with the staircase algorithm and the $QZ$ algorithm applied to
pencils derived from $S(\lambda)$. We also show how to derive a strongly
minimal system matrix from an arbitrary linear system matrix by applying to it
a reduction procedure, that only uses unitary equivalence transformations. This
implies that numerical errors performed during the reduction procedure remain
bounded. Finally, we show how to perform diagonal scalings to an arbitrary
pencil such that its row and column norms are all of the order of 1. Combined
with the fact that we use unitary transformation in both the reduction
procedure and the computation of the eigenstructure, this guarantees that we
computed the exact eigenstructure of a perturbed linear system matrix, but
where the perturbation is of the order of the machine precision.
