  Block full rank pencils introduced in [Dopico et al., Local linearizations of
rational matrices with application to rational approximations of nonlinear
eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local
information about zeros that are not poles of rational matrices. In this paper
we extend the structure of those block full rank pencils to construct
linearizations of rational matrices that allow us to recover locally not only
information about zeros but also about poles, whenever certain minimality
conditions are satisfied. In addition, the notion of degree of a rational
matrix will be used to determine the grade of the new block full rank
linearizations as linearizations at infinity. This new family of linearizations
is important as it generalizes and includes the structures appearing in most of
the linearizations for rational matrices constructed in the literature. In
particular, this theory will be applied to study the structure and the
properties of the linearizations in [P. Lietaert et al., Automatic rational
approximation and linearization of nonlinear eigenvalue problems, submitted].
