Smith forms of palindromic matrix polynomials

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

doi:10.13001/1081-3810.1426

Cited by 41 publications

(73 citation statements)

References 19 publications

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“…Sometimes, in the theory of matrix polynomials it is convenient to allow for a zero leading coefficient (see, e.g., [30]). For this reason, in our exposition we will not exclude this possibility.…”

Section: The Pencil B(x) Is Unique Up To a Permutation Of The Diagonamentioning

confidence: 99%

“…When we write about a matrix polynomial A(x) = d i=0 A i x i , we agree that the leading factor could be the zero matrix. The natural number d is therefore an arbitrarily fixed grade, equal to or larger than the degree, which is attached artificially to the polynomial [30]. However, in most applications the leading coefficient is nonzero: a reader uncomfortable with the concept of grade may simply think of d as the degree.…”

Section: The Pencil B(x) Is Unique Up To a Permutation Of The Diagonamentioning

confidence: 99%

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Duality of matrix pencils, Wong chains and linearizations

Noferini

Poloni

2015

Linear Algebra and its Applications

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We consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils. One is dual pencils, a pair of pencils with the same regular part and related singular structures. They were introduced by V. Kublanovskaya in the 1980s. The other is Wong chains, families of subspaces, associated with (possibly singular) matrix pencils, that generalize Jordan chains. They were introduced by K.T. Wong in the 1970s. Together, dual pencils and Wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications, especially in the context of linearizations of matrix polynomials.We first give a self-contained introduction to these two concepts, using modern language and extending them to a more general form; we describe the relation between them and show how they act on the Kronecker form of a pencil and on spectral and singular structures (eigenvalues, eigenvectors and minimal bases). Then we present several new applications of these results to more recent topics in matrix pencil theory, including: constraints on the minimal indices of singular Hamiltonian and symplectic pencils, new sufficient conditions under which pencils in L 1 , L 2 linearization spaces are strong linearizations, a new perspective on Fiedler pencils, and a link between the Möller-Stetter theorem and some linearizations of matrix polynomials.

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Section: The Pencil B(x) Is Unique Up To a Permutation Of The Diagonamentioning

confidence: 99%

Section: The Pencil B(x) Is Unique Up To a Permutation Of The Diagonamentioning

confidence: 99%

Duality of matrix pencils, Wong chains and linearizations

Noferini

Poloni

2015

Linear Algebra and its Applications

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“…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%

“…We also stress the importance of viewing the partial multiplicities of a fixed λ 0 as a sequence. In a number of situations, especially for matrix polynomials with structure [58,59,60], it is essential to consider certain subsequences of partial multiplicities, which can be subtly constrained by the matrix polynomial structure. Indeed, even the zeroes in the partial multiplicity sequences of structured matrix polynomials can sometimes have nontrivial significance [58,59,60].…”

Section: Definition 2 (Eigenvalues and Elementary Divisors)mentioning

confidence: 99%

“…In joint work with Mehrmann [58,59], these questions were resolved by characterizing the Smith forms for these structure classes using a novel technique based on the properties of compound matrices. This work showed that the eigenvalue pairings found in Theorem 3 do indeed extend to singular polynomials in these classes.…”

Section: Theorem 3 (Eigenvalue Pairings Of Structured Matrix Polynomimentioning

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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A permuted factors approach for the linearization of polynomial matrices

Vologiannidis

Antoniou

2011

Math. Control Signals Syst.

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new family of companion forms associated with a regular polynomial matrix T (s) has been presented, using products of permutations of n elementary matrices, generalizing similar results presented in Fiedler (Linear Algebra Its Appl 371: [325][326][327][328][329][330][331] 2003) where the scalar case was considered. In this paper, extending this "permuted factors" approach, we present a broader family of companion-like linearizations, using products of up to n(n − 1)/2 elementary matrices, where n is the degree of the polynomial matrix. Under given conditions, the proposed linearizations can be shown to consist of block entries where the coefficients of the polynomial matrix appear intact. Additionally, we provide a criterion for those linearizations to be block symmetric. We also illustrate several new block symmetric linearizations of the original polynomial matrix T (s), where in some of them the constraint of nonsingularity of the constant term and the coefficient of maximum degree are not a prerequisite.

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Smith forms of palindromic matrix polynomials

Cited by 41 publications

References 19 publications

Duality of matrix pencils, Wong chains and linearizations

Duality of matrix pencils, Wong chains and linearizations

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

A permuted factors approach for the linearization of polynomial matrices

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