Linearizations of polynomial matrices with symmetries and their applications

Abstract

In an earlier paper by the present authors, a new family of companion forms associated with a regular polynomial matrix was presented, generalizing similar results by M. Fiedler who considered the scalar case. This family of companion forms preserves both the finite and infinite elementary divisor structure of the original polynomial matrix, thus all its members can be seen as linearizations of the corresponding polynomial matrix. In this note, its applications on polynomial matrices with symmetries, which appear in a number of engineering fields, are examined.

An algorithm for the complete solution of quadratic eigenvalue problems

“…Research on linearizations of matrix polynomials has been very active in recent years, with achievements including generalization of the definition of linearization [Lancaster and Psarrakos 2005], [Lancaster 2008], derivation of new (structured) linearizations [Amiraslani et al 2009], [Antoniou and Vologiannidis 2004], [Antoniou and Vologiannidis 2006], , [Mackey et al 2006b], [Mackey et al 2006c] and analysis of the influence of the linearization process on the accuracy and stability of computed solutions , [Higham et al 2007], [Higham et al 2008].…”

Structured strong linearizations from Fiedler pencils with repetition II

“…The pencil λM z − M q is a generalized Fiedler pencil [1,2], which is known to be a strong linearization of P (λ) [5]. Therefore, we have the following result.…”

“…In the literature [2,3,8,15,16,21,22] different kinds of structured linearizations have been considered: palindromic, symmetric, skew-symmetric, alternating, etc. Among the structured linearizations, those that are strong and can be easily constructed from the coefficients of the matrix polynomial P (λ) are of particular interest [1,2,10,25,20], more precisely, those strong linearizations L P (λ) = λL 1 − L 0 such that each n × n block of L 1 and L 0 is either 0 n , ±I n , or ±A i , for i = 0, 1, ..., k, when L 1 and L 0 are viewed as k × k block matrices. There are some well-known families of linearizations with this property: Fiedler pencils ( [1], [7]), generalized Fiedler pencils ( [1], [5]), and Fiedler pencils with repetition (FPR) ( [25]).…”

“…In the literature, symmetric linearizations for symmetric matrix polynomials can be found [1,2,15,17,18,19,20,25]. We are interested in symmetric linearizations which are easily constructed from the coefficients of the matrix polynomial.…”

“…We are interested in symmetric linearizations which are easily constructed from the coefficients of the matrix polynomial. Notice that, by Lemma 5.2, for k > 1, there are no symmetric linearizations in the family of Fiedler pencils and the symmetric linearizations in the family of generalized Fiedler pencils only includes a few pencils [1,2]. Some examples of symmetric strong linearizations in the family of Fiedler pencils with repetition associated with a symmetric matrix polynomial P (λ) of degree k are given in [25], where this family is introduced.…”