Möbius transformations of matrix polynomials

Mackey, D. Steven; Mackey, Niloufer; Mehl, Christian; Mehrmann, Volker

doi:10.1016/j.laa.2014.05.013

Cited by 50 publications

(89 citation statements)

References 47 publications

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“…It was then generalized to matrix polynomials in 2006 by Mehrmann and co-authors [56], where it was shown how palindromic and alternating structures are related via a Cayley transformation of matrix polynomials. The definition of general Möbius transformations in [61] completes this development, providing an important and flexible tool for working with matrix polynomials.…”

Section: Möbius Transformationsmentioning

confidence: 91%

“…The degree of a nonzero matrix polynomial retains its usual meaning as the largest integer j such that the coefficient of λ j in P(λ ) is nonzero. The grade of a nonzero matrix polynomial is a choice of integer k at least as large as its degree [22,59,61]. It signals that the polynomial is to be viewed as an element of a particular vector space -the F-vector space of all matrix polynomials of degree less than or equal to k. Choosing a grade, in effect, specifies the finite-dimensional vector space of discourse.…”

Section: Basic Conceptsmentioning

confidence: 99%

“…A useful investigative tool developed by Mehrmann and his co-authors in the last few years is the extension of linear fractional rational transformations (i.e., Möbius transformations) to transformations that act on matrix polynomials [61]. One of the main motivations for this work is understanding the relationships between different classes of structured matrix polynomials.…”

Section: Möbius Transformationsmentioning

confidence: 99%

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Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Mackey

Tisseur

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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Matrix polynomial eigenproblems arise in many application areas, both directly and as approximations for more general nonlinear eigenproblems. One of the most common strategies for solving a polynomial eigenproblem is via a linearization, which replaces the matrix polynomial by a matrix pencil with the same spectrum, and then computes with the pencil. Many matrix polynomials arising from applications have additional algebraic structure, leading to symmetries in the spectrum that are important for any computational method to respect. Thus it is useful to employ a structured linearization for a matrix polynomial with structure. This essay surveys the progress over the last decade in our understanding of linearizations and their construction, both with and without structure, and the impact this has had on numerical practice.

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Section: Möbius Transformationsmentioning

confidence: 91%

Section: Basic Conceptsmentioning

confidence: 99%

Section: Möbius Transformationsmentioning

confidence: 99%

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Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Mackey

Tisseur

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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“…. , P s (λ) in Lemma 2.8, although this strategy changes the degree of the elementary divisors at infinity of each P i (λ) with degree smaller than the grade by a uniform shift [3,14].…”

Section: Preliminariesmentioning

confidence: 99%

“…. , w r (λ) forms a minimal basis, it is known (see [2,Thm 3.2] or [14,Thm 7.5]) that rev w 1 (λ), . .…”

Section: Preliminariesmentioning

confidence: 99%

Polynomial zigzag matrices, dual minimal bases, and the realization of completely singular polynomials

Terán

Dopico

Mackey

et al. 2016

Linear Algebra and its Applications

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Minimal bases of rational vector spaces are a well known and important tool in systems theory. If minimal bases for two subspaces of rational n-space are displayed as the rows of polynomial matrices Z1(λ) k×n and Z2(λ)m×n, respectively, then Z1 and Z2 are said to be dual minimal bases if the subspaces have complementary dimension, i.e., k + m = n, and Z 1 (λ)Z T 2 (λ) = 0. In other words, each Zj(λ) provides a minimal basis for the nullspace of the other. It has long been known that for any dual minimal bases Z1(λ) and Z2(λ), the row degree sums of Z1 and Z2 are the same. In this paper we show that this is the only constraint on the row degrees, thus characterizing the possible row degrees of dual minimal bases. The proof is constructive, making extensive use of a new class of sparse, structured polynomial matrices that we have baptized zigzag matrices. Another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices -given a list of left and right minimal indices and a desired degree d, does there exist a completely singular matrix polynomial (i.e., a matrix polynomial with no elementary divisors whatsoever) of degree d having exactly the prescribed minimal indices? We show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices. The constructed realization is simple, and explicitly displays the desired minimal indices in a fashion analogous to the classical Kronecker canonical form of singular pencils.

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Block Kronecker linearizations of matrix polynomials and their backward errors

et al. 2018

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We introduce a new family of strong linearizations of matrix polynomials-which we call "block Kronecker pencils"-and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This stability analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. The global backward error analysis in this work presents for the first time the following key properties: it is a rigurous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of "strong block minimal bases pencils", which are robust under certain perturbations and, so, include certain perturbations of block Kronecker pencils. We hope that this robustness property will allow us to extend the results in this paper to other contexts.

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Möbius transformations of matrix polynomials

Cited by 50 publications

References 47 publications

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Polynomial zigzag matrices, dual minimal bases, and the realization of completely singular polynomials

Block Kronecker linearizations of matrix polynomials and their backward errors

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