Triangularizing matrix polynomials

Taslaman, Leo; Tisseur, Françoise; Zaballa, Ion

doi:10.1016/j.laa.2013.05.006

Cited by 21 publications

(34 citation statements)

References 10 publications

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“…The result of Theorem 9.4 is also presented in [38] and [61], and extended in each paper to an even more general result. We have included the regular realization theorem and an alternative proof in this paper primarily to highlight the importance of Möbius transformations as a tool for the solution of polynomial inverse eigenproblems.…”

Section: Theorem 94 (Regular Realization Theorem)mentioning

confidence: 79%

Möbius transformations of matrix polynomials

Mackey

Mehl

et al. 2015

Linear Algebra and its Applications

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We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated.

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Section: Theorem 94 (Regular Realization Theorem)mentioning

confidence: 79%

Möbius transformations of matrix polynomials

Mackey

Mehl

et al. 2015

Linear Algebra and its Applications

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“…Moreover, we propose an adaptation of the moment method to the computation of solvents. Finally, we build on existing work on triangularization of matrix polynomials (see [38] and [35]) and explore the relationship between solvents of matrix polynomials in general and in triangularized form.…”

Section: P (S) := J=0mentioning

confidence: 99%

“…Motivated by the results in [35] and [38], where the authors analyze a method for triangularizing the matrix polynomial P (λ), we aim here to study the relation between solvents of general and of triangularized matrix polynomials.…”

Section: Solvents and Triangularized Matrix Polynomialsmentioning

confidence: 99%

“…For any algebraically closed field F, any matrix polynomial P (λ) ∈ F[λ] n×m , with n ≤ m, can be reduced to triangular form via unimodular transformations, preserving the degree and the finite and infinite elementary divisors [35], [38]. What can we say about solvents for a given matrix polynomial P (λ) and for the associated triangularized polynomial?…”

Section: Triangularizing Matrix Polynomialsmentioning

confidence: 99%

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A contour integral approach to the computation of invariant pairs

Barkatou

Boito

Ugalde

2017

Theoretical Computer Science

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We study some aspects of the invariant pair problem for matrix polynomials, as introduced by Betcke and Kressner [3] and by Beyn and Thümmler [6]. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. We compute formulations for the condition numbers and the backward error for invariant pairs and solvents. We then adapt the Sakurai-Sugiura moment method [1] to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues. Numerical refinement via a variant of Newton's method is also studied. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.

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“…A matrix polynomial U (λ) ∈ F [λ] n×n is said to be unimodular if det U (λ) ∈ F \ {0}, and two matrix polynomials that differ only by multiplication by unimodular matrix polynomials (from the left and the right) are said to be equivalent. It was shown in [16] and [17] that unimodular transformations are enough to reduce any square matrix polynomial to triangular form over C and quasi-triangular form over R, while preserving the degree. Of course, this includes the case of Hessenberg form since (quasi-) triangular matrices are also Hessenberg.…”

Section: Introduction Almost All Matrices In Cmentioning

confidence: 99%

Reduction of Matrix Polynomials to Simpler Forms

Nakatsukasa¹,

Taslaman²,

Tisseur³

et al. 2018

SIAM J. Matrix Anal. & Appl.

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Abstract. A square matrix can be reduced to simpler form via similarity transformations. Here "simpler form" may refer to diagonal (when possible), triangular (Schur), or Hessenberg form. Similar reductions exist for matrix pencils if we consider general equivalence transformations instead of similarity transformations. For both matrices and matrix pencils, well-established algorithms are available for each reduction, which are useful in various applications. For matrix polynomials, unimodular transformations can be used to achieve the reduced forms but we do not have a practical way to compute them. In this work we introduce a practical means to reduce a matrix polynomial with nonsingular leading coefficient to a simpler (diagonal, triangular, Hessenberg) form while preserving the degree and the eigenstructure. The key to our approach is to work with structure preserving similarity transformations applied to a linearization of the matrix polynomial instead of unimodular transformations applied directly to the matrix polynomial. As an application, we illustrate how to use these reduced forms to solve parameterized linear systems.

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Triangularizing matrix polynomials

Cited by 21 publications

References 10 publications

Möbius transformations of matrix polynomials

Möbius transformations of matrix polynomials

A contour integral approach to the computation of invariant pairs

Reduction of Matrix Polynomials to Simpler Forms

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