  Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian,
skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many
applications. Linearizations of rational matrices have been introduced recently
for computing poles, eigenvalues, eigenvectors, minimal bases and minimal
indices of rational matrices. For structured rational matrices, it is desirable
to construct structure-preserving linearizations so as to preserve the symmetry
in the eigenvalues and poles of the rational matrices. With a view to
constructing structure-preserving linearizations of structured rational
matrices, we propose a family of Fiedler-like pencils and show that the family
of Fiedler-like pencils is a rich source of structure-preserving strong
linearizations of structured rational matrices. We construct symmetric,
skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian,
para-Hermitian and para-skew-Hermitian strong linearizations of a rational
matrix $G(\lambda)$ when $G(\lambda)$ has the same structure. Further, when
$G(\lambda)$ is real and symmetric, we show that the transfer functions of real
symmetric linearizations of $G(\lambda)$ preserve the Cauchy-Maslov index of
$G(\lambda).$ We describe the recovery of eigenvectors, minimal bases and
minimal indices of $G(\lambda)$ from those of the linearizations of
$G(\lambda)$ and show that the recovery is operation-free.
