Large vector spaces of block-symmetric strong linearizations of matrix polynomials

Abstract

Abstract. Given a matrix polynomial P (λ) = P k i=0 λ i A i of degree k, where A i are n × n matrices with entries in a field F, the development of linearizations of P (λ) that preserve whatever structure P (λ) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P (λ) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P (λ) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space called DL(P ) of dimension k of block-symmetric pencils such that most of them are linearizations, which form an infinite family when the field F is infinite. One drawback of the pencils in DL(P ) is that none of them is a linearization when P (λ) is singular. In this paper we introduce many new classes of block-symmetric linearizations for P (λ), when k ≥ 3, which have three key features: (a) their constructions are based on an new class of Fiedler-like pencils that we call generalized Fiedler pencils with repetition; (b) they are defined in terms of vector spaces of dimension O(n 2 ) of block-symmetric pencils most of which are linearizations; and (c) when k is odd, many of these classes contain linearizations even when P (λ) is singular. Therefore, the two fundamental contributions of this manuscript are that the dimensions of the new spaces of block-symmetric linearizations are much larger than the dimension of DL(P ) (for n ≥ √ k) and that vector spaces of block-symmetric linearizations valid for singular matrix polynomials are presented for the first time.In particular, the largest dimension of the subspaces of block-symmetric pencils we introduce is jand the coefficients of the pencils in the new subspaces can be easily constructed as k × k block-matrices whose n × n blocks are of the form 0, αIn, ±αA i , or arbitrary n × n matrices, where α is an arbitrary nonzero scalar.

Linearizations of Hermitian Matrix Polynomials Preserving the Sign Characteristic

“…In [10], the family of GFPR associated with a matrix polynomial P ( ) of degree k as in (2.1) was introduced as an extension of the family of Fiedler pencils with repetition (FPR) presented in [32].…”

“…In addition, for practical purposes, it is essential that these linearizations are easily constructible from the coefficients of the polynomial. References [3,7,10,11,12,19,21,22,32] are a small sample of papers where new classes of linearizations have been presented.…”

“…Infinitely many other linearizations that preserve the sign characteristic of P ( ) are presented in Section 7. They belong to the family of generalized Fiedler pencils with repetition (GFPR), introduced very recently in [10], which extends in a nontrivial way the set of pencils forming the standard basis of DL(P ). Particularly simple examples of these GFPR linearizations are carefully described in Section 8 (see Example 8.2).…”

“…The most relevant linearizations of n ⇥ n matrix polynomials of grade k used in practice are easily described when their coefficients are viewed as block-matrices partitioned into k ⇥ k blocks of size n ⇥ n [3,7,10,11,12,17,19,21,22,32]. The description of some structures of linearizations requires the definition of blocktransposition and block-symmetry.…”

Fiedler-comrade and Fiedler--Chebyshev pencils

“…Since the structure of a matrix polynomial is reflected in its spectrum, numerical methods to solve polynomial eigenvalue problems should exploit to a maximal extent the structure of matrix polynomials [21]. For this reason, finding linearizations that retain whatever structure the matrix polynomial P (x) might possess is a fundamental problem in the theory of linearizations (see, for example, [4,5,8,21] and the references therein). The results in this work expand the arena in which to look for linearizations of matrix polynomials expressed in some orthogonal polynomial bases having additional useful properties.…”