  The aim of this paper is twofold. First, we introduce a new class of
linearizations, based on the generalization of a construction used in
polynomial algebra to find the zeros of a system of (scalar) polynomial
equations. We show that one specific linearization in this class, which is
constructed naturally from the QR factorization of the matrix obtained by
stacking the coefficients of $A(x)$, has good conditioning and stability
properties.
  Moreover, while analyzing this class, we introduce a general technique to
derive new linearizations from existing ones. This technique generalizes some
ad-hoc arguments used in dealing with the existing linearization classes, and
can hopefully be used to derive a simpler and more general theory of
linearizations. This technique relates linearizations to \emph{pencil
arithmetic}, a technique used in solving matrix equations that allows to extend
some algebraic operations from matrix to matrix pencils.
