On vector spaces of linearizations for matrix polynomials in orthogonal bases

Abstract

Regular and singular matrix polynomials P (λ) = k i=0 P i φ i (λ), P i ∈ R n×n given in an orthogonal basis φ 0 (λ), φ 1 (λ), . . . , φ k (λ) are considered. Following the ideas in [9], the vector spaces, called M 1 (P ), M 2 (P ) and DM(P ), of potential linearizations for P (λ) are analyzed. All pencils in M 1 (P ) are characterized concisely. Moreover, several easy to check criteria whether a pencil in M 1 (P ) is a (strong) linearization of P (λ) are given. The equivalence of some of them to the Z-rank-condition [9] is pointed out. Results on the vector space dimensions, the genericity of linearizations in M 1 (P ) and the form of block-symmetric pencils are derived in a new way on a basic algebraic level. Moreover, an extension of these results to degree-graded bases is presented. Throughout the paper, structural resemblances between the matrix pencils in L 1 , i.e. the results obtained in [9], and their generalized versions are pointed out.

Strong linearizations of rational matrices with polynomial part expressed in an orthogonal basis

“…In this section and in Section 4 we present strong linearizations of square rational matrices G(λ) with polynomial part D(λ) expressed in an orthogonal basis. More precisely, we consider strong linearizations of D(λ) that belong to the ansatz spaces M 1 (D) or M 2 (D), recently developed by H. Faßbender and P. Saltenberger in [15], and based on them, we construct strong linearizations of G(λ) by using Lemma 2.7 and the strong linearizations presented in [4,Section 8.2].…”

“…As said in the preliminaries, we consider an arbitrary field F throughout this paper, although the results in [15] are stated only for the real field R. Nevertheless, the results of [15] that are used in this paper are also valid for any field F. We consider a polynomial basis {φ j (λ)} ∞ j=0 of F[λ], viewed as an F-vector space, with φ j (λ) a polynomial of degree j, that satisfies the following three-term recurrence relation:…”

“…For answering the question posed in the previous paragraph, we construct in this paper strong linearizations of a rational matrix by using strong linearizations of its polynomial part D(λ) that belong to the other important family of strong linearizations of polynomial matrices (which are not strong block minimal bases pencils in general), i.e., the so-called vector spaces of linearizations, originally introduced in [26], further studied in [10,28], and recently extended in [15]. In particular, we consider in this paper strong linearizations of D(λ) that belong to the ansatz spaces M 1 (D) or M 2 (D), developed by Faßbender and Saltenberger in [15]. Therefore, the results in this paper are of interest when rational matrices with nontrivial polynomial part are considered, that is, rational matrices in which the polynomial part has degree greater than or equal to two.…”

“…In Section 5, we show how to recover the eigenvectors of the rational matrix from those of its strong linearizations constructed in Sections 3 and 4. Moreover, given a symmetric rational matrix, in Section 7 we construct strong linearizations that preserve its symmetric structure by using symmetric realizations of the strictly proper part, which are introduced in Section 6, and strong linearizations in the double ansatz space DM(D) [15] of the polynomial part. In Section 8, we present analogous results for Hermitian rational matrices.…”

Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

“…It was shown in [5] that even when P(λ ) is square but singular, almost every pencil in L 1 (P) and L 2 (P) is a linearization of P(λ ) from which the solution of complete eigenvalue problem for P(λ ) can be easily recovered. The vector space setting for constructing linearizations has since been extended to cover other polynomial bases [13] and inspired further work that throws fresh light on these spaces [23]. Other important choices of linearizations not covered by L 1 (P) and L 2 (P) are the Fiedler pencils and their generalizations [1,29,6,2,4] which are also sources of linearizations for non-square matrix polynomials [7].…”