 
Summary: RECOVERY OF EIGENVECTORS AND MINIMAL BASES OF MATRIX
POLYNOMIALS FROM GENERALIZED FIEDLER LINEARIZATIONS
MARÍA I. BUENO, FERNANDO DE TERÁN , AND FROILÁN M. DOPICO §
Abstract. A standard way to solve polynomial eigenvalue problems P()x = 0 is to convert the matrix
polynomial P() into a matrix pencil that preserves its elementary divisors and, therefore, its eigenvalues. This
process is known as linearization and is not unique, since there are infinitely many linearizations with widely varying
properties associated with P(). This freedom has motivated the recent development and analysis of new classes
of linearizations that generalize the classical first and second Frobenius companion forms, with the goals of finding
linearizations that retain whatever structures that P() might possess and/or of improving numerical properties,
as conditioning or backward errors, with respect the companion forms. In this context, an important new class of
linearizations is what we name generalized Fiedler linearizations, introduced in 2004 by Antoniou and Vologiannidis
as an extension of certain linearizations introduced previously by Fiedler for scalar polynomials. On the other hand,
the mere definition of linearization does not imply the existence of simple relationships between the eigenvectors,
minimal indices, and minimal bases of P() and those of the linearization. So, given a class of linearizations, to
provide easy recovery procedures for eigenvectors, minimal indices, and minimal bases of P() from those of the
linearizations is essential for the usefulness of this class. In this paper we develop such recovery procedures for
generalized Fiedler linearizations and pay special attention to structure preserving linearizations inside this class.
Key words. eigenvector, Fiedler pencils, linearizations, matrix polynomials, minimal bases, minimal indices,
palindromic polynomials, symmetric polynomials
AMS subject classifications. 65F15, 15A18, 15A22.
