Fiedler-Comrade and Fiedler-Chebyshev pencils

Vanni Noferini, Javier Perez Alvaro

    Research output: Contribution to journalArticlepeer-review


    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 formulas for eigenvectors, and minimal indices and bases.

    Original languageEnglish
    Pages (from-to)1600-1624
    Number of pages25
    JournalSIAM Journal on Matrix Analysis and Applications
    Issue number4
    Early online date8 Nov 2016
    Publication statusPublished - 2016


    • Chebyshev polynomial
    • Eigenvector recovery
    • Fiedler pencil
    • Linearization
    • Matrix polynomial
    • Minimal basis
    • Minimal indices
    • Singular matrix polynomial


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