  The standard way of solving the polynomial eigenvalue problem associated with
a matrix polynomial is to embed the matrix polynomial into a matrix pencil,
transforming the problem into an equivalent generalized eigenvalue problem.
Such pencils are known as linearizations. Many of the families of
linearizations for matrix polynomials available in the literature are
extensions of the so-called family of Fiedler pencils. These families are known
as generalized Fiedler pencils, Fiedler pencils with repetition and generalized
Fiedler pencils with repetition, or Fiedler-like pencils for simplicity. The
goal of this work is to unify the Fiedler-like pencils approach with the more
recent one based on strong block minimal bases pencils introduced in
\cite{canonical}. To this end, we introduce a family of pencils that we have
named extended block Kronecker pencils, whose members are, under some generic
nonsingularity conditions, strong block minimal bases pencils, and show that,
with the exception of the non proper generalized Fiedler pencils, all
Fiedler-like pencils belong to this family modulo permutations. As a
consequence of this result, we obtain a much simpler theory for Fiedler-like
pencils than the one available so far. Moreover, we expect this unification to
allow for further developments in the theory of Fiedler-like pencils such as
global or local backward error analyses and eigenvalue conditioning analyses of
polynomial eigenvalue problems solved via Fiedler-like linearizations.
