The quasi-Weierstraß form for regular matrix pencils

Berger, Toby; Ilchmann, Achim; Trenn, Stephan

doi:10.1016/j.laa.2009.12.036

Cited by 143 publications

(164 citation statements)

References 16 publications

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“…This characterization seems to be implicit in results such as [4,Section 3] and [36, Section 2], but it helps to state it in full form with proofs.…”

Section: Theorem 44 the Subspacesmentioning

confidence: 98%

“…In [4,5,40] only the special cases λ = ∞, µ = 0 and λ = 0, µ = ∞ appear, while in [6,36] the authors allow also λ ∈ C, µ = ∞. Theorem 4.4 implies that it is possible to change the second base point µ without altering the corresponding subspace chain.…”

Section: Theorem 44 the Subspacesmentioning

confidence: 99%

“…Wong chains have been introduced in [40], and only recently reappeared in the study of matrix pencils [4,5,6]; the definition that we use here is a generalized version. Dual pencils appear in [26,Section 1.3], where they are given the name of consistent pencils and some of their theoretical properties are stated.…”

Section: Introductionmentioning

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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“…This characterization seems to be implicit in results such as [4,Section 3] and [36, Section 2], but it helps to state it in full form with proofs.…”

Section: Theorem 44 the Subspacesmentioning

confidence: 98%

Section: Theorem 44 the Subspacesmentioning

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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“…Recently, see [7,8,9], Wong sequences are used to prove the KCF and compared to the proof by Gantmacher [15] they provide some geometrical insight. The Wong sequences have their origin in Wong [27].…”

Section: Wong Sequencesmentioning

confidence: 99%

Linear relations and the Kronecker canonical form

Berger

Trunk

Winkler

2016

Linear Algebra and its Applications

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Abstract. We show that the Kronecker canonical form (which is a canonical decomposition for pairs of matrices) is the representation of a linear relation in a finite dimensional space. This provides a new geometric view upon the Kronecker canonical form. Each of the four entries of the Kronecker canonical form has a natural meaning for the linear relation which it represents. These four entries represent the Jordan chains at finite eigenvalues, the Jordan chains at infinity, the so-called singular chains and the multi-shift part. Or, to state it more concise: For linear relations the Kronecker canonical form is the analogue of the Jordan canonical form for matrices.

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“…Following [1] the decoupling (4) is called quasi-Weierstrass form (QWF), which can easily obtained via the so-called Wong sequences.…”

Section: Dae Preliminariesmentioning

confidence: 99%

Stabilization of switched DAEs via fast switching

Trenn

2016

Proc Appl Math and Mech

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Switched differential algebraic equations (switched DAEs) can model dynamical systems with state constraints together with sudden structural changes (switches). These switches may lead to induced jumps and can destabilize the system even in the case that each mode is stable. However, the opposite effect is also possible; in particular, the question of finding a stabilizing switching signal is of interest. Two approaches are presented how to stabilize a switched DAE via fast switching.

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The quasi-Weierstraß form for regular matrix pencils

Cited by 143 publications

References 16 publications

Duality of matrix pencils, Wong chains and linearizations

Duality of matrix pencils, Wong chains and linearizations

Linear relations and the Kronecker canonical form

Stabilization of switched DAEs via fast switching

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