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The standard way of solving the polynomial eigenvalue problem associated witha 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 oflinearizations for matrix polynomials available in the literature areextensions of the so-called family of Fiedler pencils. These families are knownas generalized Fiedler pencils, Fiedler pencils with repetition and generalizedFiedler pencils with repetition, or Fiedler-like pencils for simplicity. Thegoal of this work is to unify the Fiedler-like pencils approach with the morerecent one based on strong block minimal bases pencils introduced in\cite{canonical}. To this end, we introduce a family of pencils that we havenamed extended block Kronecker pencils, whose members are, under some genericnonsingularity conditions, strong block minimal bases pencils, and show that,with the exception of the non proper generalized Fiedler pencils, allFiedler-like pencils belong to this family modulo permutations. As aconsequence of this result, we obtain a much simpler theory for Fiedler-likepencils than the one available so far. Moreover, we expect this unification toallow for further developments in the theory of Fiedler-like pencils such asglobal or local backward error analyses and eigenvalue conditioning analyses ofpolynomial eigenvalue problems solved via Fiedler-like linearizations.

A unified approach to Fiedler-like pencils via strong block minimal bases pencils