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 [15]. 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.Key words. Fiedler pencils, generalized Fiedler pencils, Fiedler pencils with repetition, generalized Fiedler pencils with repetition, matrix polynomials, strong linearizations, block minimal bases pencils, block Kronecker pencils, extended block Kronecker pencils, minimal basis, dual minimal bases AMS subject classifications. 65F15, 15A18, 15A22, 15A54 1. Introduction. Matrix polynomials and their associated polynomial eigenvalue problems appear in many areas of applied mathematics, and they have received in the last years considerable attention. For example, they are ubiquitous in a wide range of problems in engineering, mechanic, control theory, computer-aided graphic design, etc. For detailed discussions of different applications of matrix polynomials, we refer the reader to the classical references [23,28,42], the modern surveys [2, Chapter 12] and [37,43] (and their references therein), and the references [32,33,34]. For those readers not familiar with the theory of matrix polynomials and polynomial eigenvalue problems, those topics are briefly reviewed in Section 2.The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to linearize the polynomial into a matrix pencil (i.e., matrix polynomials of grade 1), known as linearization [13,22,23]. The linearization process transforms the polynomial eigenvalue problem into an equivalent generalized eigenvalue problem, which, then, can be solved using mature and well-understood eigensolvers such as the QZ algorithm or the
Using the transfer principle, we classify the periodic points on the regular n-gon and double n-gon translation surfaces and deduce consequences for the finite blocking problem on rational triangles that unfold to these surfaces.
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