Abstract. In many applications, the polynomial eigenvalue problem, P (λ)x = 0, arises with P (λ) being a structured matrix polynomial (for example, palindromic, symmetric, skew-symmetric). In order to solve a structured polynomial eigenvalue problem it is convenient to use strong linearizations with the same structure as P (λ) to ensure that the symmetries in the eigenvalues due to that structure are preserved in numerical computations. In this paper we characterize all the pencils in the family of the Fiedler pencils with repetition, introduced by Vologiannidis and Antoniou [25], associated with a square matrix polynomial P (λ) that are block-symmetric for every matrix polynomial P (λ). We show that this family of pencils is precisely the set of all Fiedler pencils with repetition that are symmetric when P (λ) is. When some generic nonsingularity conditions are satisfied, these pencils are strong linearizations of P (λ). In particular, our family strictly contains the standard basis for DL(P ), a k-dimensional vector space of symmetric pencils introduced by Mackey, Mackey, Mehl, and Mehrmann [20].
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.
Abstract. The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such linearizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In this context, Hermitian matrix polynomials are one of the most important classes of matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic is a set of signs attached to these real eigenvalues which is crucial for determining the behavior of systems described by Hermitian matrix polynomials and, therefore, it is desirable to develop linearizations that preserve the sign characteristic of these polynomials, but, at present, only one such linearization is known. In this paper, we present a complete characterization of all the Hermitian strong linearizations that preserve the sign characteristic of a given Hermitian matrix polynomial and identify several families of such linearizations that can be constructed very easily from the coefficients of the polynomial.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.