Geometry of Matrix Polynomial Spaces

Dmytryshyn, Andrii; Johansson, Stefan; Kågström, Bo; Dooren, Paul Van

doi:10.1007/s10208-019-09423-1

Cited by 10 publications

(20 citation statements)

References 36 publications

(97 reference statements)

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“…To be exact, we find such a perturbation of the matrix polynomial coefficients that the linearization of this perturbed polynomial (2.2) has the same complete eigenstructure as a given perturbed linearization (2.1). We also note that the existence of such a perturbation (2.2) was proven before for Fiedler-type linearizations [8,14,32], and even for a larger class of block-Kronecker linearizations [15]. Nevertheless, the problem of finding this perturbation explicitly has been open until now.…”

Section: Introductionmentioning

confidence: 75%

“…Nonlinear eigenvalue problems play an important role in mathematics and its applications, see e.g., the surveys [21,26,30]. In particular, polynomial eigenvalue problems have been receiving much attention [3,14,15,22,24,25]. Recall that…”

Section: Introductionmentioning

confidence: 99%

“…Frequently, complete eigenstructures, i.e. elementary divisors and minimal indices of matrix polynomials (for the definitions, see e.g., [6,14]) provide an understanding of the properties and behaviour of the underlying physical system and thus are the actual objects of interest. The complete eigenstructure is usually computed by passing to a (strong) linearization which replaces a matrix polynomial by a matrix pencil, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…One example is solving various distance problems for matrix polynomials if the corresponding problems are solved for matrix pencils, e.g., finding a singular matrix polynomial nearby a given matrix polynomial [4,18,20]. Another application lies in the stratification theory [8,14]: constructing an explicit perturbation of a matrix polynomial when a perturbation of its linearization is known. This will allow to say which perturbation does a certain change to the complete eigenstructure of a given polynomial.…”

Section: Introductionmentioning

confidence: 99%

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Recovering a perturbation of a matrix polynomial from a perturbation of its first companion linearization

Dmytryshyn

2021

Bit Numer Math

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A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.

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Section: Introductionmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Recovering a perturbation of a matrix polynomial from a perturbation of its first companion linearization

Dmytryshyn

2021

Bit Numer Math

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“…Note that B s (P ) is an infinite union (over all the possible values of the distinct eigenvalues) of the orbits of the polynomials whose complete eigenstructures differ from the one of P (λ) only in the values of the distinct eigenvalues. Analogous definitions for general and skew-symmetric matrix polynomials are given in [16,17,18,21]. It is important to emphasize that, unlike what happens for matrix pencils, two symmetric matrix polynomials with the same size, grade, and complete eigenstructure are not necessarily congruent.…”

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confidence: 99%

Generic Symmetric Matrix Polynomials with Bounded Rank and Fixed Odd Grade

Terán¹,

Dmytryshyn²,

Dopico³

2020

SIAM J. Matrix Anal. Appl.

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We determine the generic complete eigenstructures for n × n complex symmetric matrix polynomials of odd grade d and rank at most r. More precisely, we show that the set of n × n complex symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and rank at most r is the union of the closures of the ⌊rd 2⌋ + 1 sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then, we prove that these sets are open in the set of n × n complex symmetric matrix polynomials of odd grade d and rank at most r. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure, in the case where all their elementary divisors are different from each other and of degree 1. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than one, because the symmetric ones include eigenvalues while the others not. This difference requires to use new techniques.

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