Addition to “The Quasi-Kronecker Form for Matrix Pencils''

Berger, Toby; Trenn, Stephan

doi:10.1137/120883244

Cited by 38 publications

(48 citation statements)

References 30 publications

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“…The characterization of Wong chains given here provides an immediate explanation of the following fact, already showed in [6]: the right minimal indices, which are well defined, can be determined from the fact that, for any λ which is not an eigenvalue of…”

Section: Regular Kronecker Blocks) Then the Wong Chains Of B(x) Are mentioning

confidence: 69%

“…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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Section: Regular Kronecker Blocks) Then the Wong Chains Of B(x) Are mentioning

confidence: 69%

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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“…Kronecker proved in [17] that any pair of matrices F , G can be transformed into a canonical form, see also [8,9,15]. Here we refer to the version in [15].…”

Section: Kronecker Canonical Formmentioning

confidence: 99%

“…[8,9,21,22,23], where the entries of α are the orders of the infinite elementary divisors, the entries of β are the column minimal indices and the entries of γ are the row minimal indices.…”

Section: Theorem 42 (Kronecker Canonical Form)mentioning

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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Controlled invariance for DAEs

Berger

2015

Proc Appl Math and Mech

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We study the concept of locally controlled invariant submanifolds for nonlinear descriptor systems. In contrast to classical approaches, we define controlled invariance as the property of solution trajectories to evolve in a given submanifold whenever they start in it. It is then shown that this concept is equivalent to the existence of a feedback which renders the closed‐loop vector field invariant in the descriptor sense. This result is motivated by a preliminary consideration of the linear case. Local controlled invariance leads to the concept of output zeroing submanifolds. We show that the outcome of the differential‐algebraic version of the zero dynamics algorithm yields a locally maximal output zeroing submanifold. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Addition to “The Quasi-Kronecker Form for Matrix Pencils''

Cited by 38 publications

References 30 publications

Duality of matrix pencils, Wong chains and linearizations

Duality of matrix pencils, Wong chains and linearizations

Linear relations and the Kronecker canonical form

Controlled invariance for DAEs

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