Fiedler-comrade and Fiedler--Chebyshev pencils

Abstract

Abstract. Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive Fiedler-comrade pencils for two bases that are extremely important in practical applications: the Chebyshev polynomials of the first and second kind. The new approach allows one to construct linearizations having limited bandwidth: a Chebyshev analogue of the pentadiagonal Fiedler pencils in the monomial basis. Moreover, our theory allows for linearizations of square matrix polynomials expressed in the Chebyshev basis (and in other bases), regardless of whether the matrix polynomial is regular or singular, and for recovery formulae for eigenvectors, and minimal indices and bases.

Root polynomials and their role in the theory of matrix polynomials

“…First, to establish rigorously and with detail the definition and the most important properties of root polynomials in the general setting of singular matrix polynomials. Second, to show that root polynomials interact naturally with a number of problems that have attracted the attention of the research community in the last years as, for instance, rational transformations of matrix polynomials, with particular emphasis on Möbius transformations, [15,17,18], linearizations of matrix polynomials and related recovery properties [1,2,4,5,14,16,19],and dual pencils [20]. We hope that the third goal of this paper will be obtained as a result of the two previous ones, since we expect that our manuscript will encourage the research community to familiarize with, and to use more often, root polynomials, which should be, in our opinion, one of the fundamental tools of any researcher on the theory of matrix polynomials.…”

Linearizations of matrix polynomials in Newton bases

“…These Newton-Fiedler pencils now constitute a third successful adaptation of Fiedler pencils to matrix polynomials expressed in a non-standard basis; the first two were to polynomials in Bernstein bases [25], and to polynomials in Chebyshev bases [28]. We have shown that all Newton-Fiedler pencils are strong linearizations for square matrix polynomials over arbitrary fields.…”

Block minimal bases ℓ-ifications of matrix polynomials

“…The above list is just a sample of linearizations, quadratifications, and ℓ-ifications given in order to show that a great part of the recent work on ℓ-ifications (linearizations, quadratifications, etc) is included in the block minimal bases matrix polynomials framework. Many other constructions fit also in this framework [45,46,48].…”