  We prove that we can always construct strongly minimal linearizations of an
arbitrary rational matrix from its Laurent expansion around the point at
infinity, which happens to be the case for polynomial matrices expressed in the
monomial basis. If the rational matrix has a particular self-conjugate
structure we show how to construct strongly minimal linearizations that
preserve it. The structures that are considered are the Hermitian and
skew-Hermitian rational matrices with respect to the real line, and the
para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary
axis. We pay special attention to the construction of strongly minimal
linearizations for the particular case of structured polynomial matrices. The
proposed constructions lead to efficient numerical algorithms for constructing
strongly minimal linearizations. The fact that they are valid for {\em any}
rational matrix is an improvement on any other previous approach for
constructing other classes of structure preserving linearizations, which are
not valid for any structured rational or polynomial matrix. The use of the
recent concept of strongly minimal linearization is the key for getting such
generality.
