Miniversal deformations of pairs of skew-symmetric matrices under congruence

Dmytryshyn, Andrii

doi:10.1016/j.laa.2016.06.015

Cited by 11 publications

(26 citation statements)

References 29 publications

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“…The codimensions of the congruence orbits of skewsymmetric matrix pencils are obtained from the solutions of the associated homogeneous systems of matrix equations in [14] (they can also be obtained by computing the numbers of independent parameters in the miniversal deformations [8]). The Matrix Canonical Structure toolbox for MATLAB was extended by the functions for calculating these codimensions [13].…”

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confidence: 99%

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

Dmytryshyn

Kågström

2014

SIAM J. Matrix Anal. & Appl.

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Abstract. We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skew-symmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another skew-symmetric matrix pencil. The developed theory relies on our main theorem stating that a skew-symmetric matrix pencil A − λB can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C − λD if and only if A − λB can be approximated by pencils congruent to C − λD.Key words. skew-symmetric matrix pencil, stratification, canonical structure information, orbit, bundle AMS subject classifications. 15A21, 15A22, 65F15, 47A07 DOI. 10.1137/1409568411. Introduction. How canonical information changes under perturbations, e.g., the confluence and splitting of eigenvalues of a matrix pencil, are essential issues for understanding and predicting the behavior of the physical system described by the matrix pencil. In general, these problems are known to be ill-posed: small perturbations in the input data may lead to drastical changes in the answers. The ill-posedness stems from the fact that both the canonical forms and the associated reduction transformations are discontinuous functions of the entries of A − λB. Therefore it is important to get knowledge about the canonical forms (or canonical structure information) of the pencils that are close to A − λB. One way to investigate this problem is to construct the stratification (i.e., the closure hierarchy) of orbits and bundles of the pencils [17].The stratification of matrix pencils under strict equivalence transformations [16,17,18] as well as the stratification of controllability and observability pairs [19] are known. StratiGraph [21,24] is a software tool for computing and visualization of such stratifications. The stratification of full normal rank matrix polynomials has been studied [22] and implemented in StratiGraph too (available as a prototype now).Our objective is to stratify orbits and bundles of skew-symmetric matrix pencils, i.e., A − λB with A T = −A and B T = −B, under congruence transformations. Skew-symmetric matrix pencils appear in several applications, e.g., the design of a passive velocity field controller [23], multisymplectic partial differential equations [3], and systems with bi-Hamiltonial structure [27]. Structure preserving linearizations of skew-symmetric matrix polynomials [25] allow us to investigate differential algebraic systems of higher orders via skew-symmetric matrix pencils. Canonical forms of skew-symmetric matrix pencils [29,30] and the structured staircase algorithm [2,4]

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confidence: 99%

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

Dmytryshyn

Kågström

2014

SIAM J. Matrix Anal. & Appl.

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“…that is congruent to (A, B), in which A is a nonsingular matrix and each (A i , B i ) is of the form J n or L n (see (2) and (3)).…”

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confidence: 99%

Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

Bovdi

Gerasimova

Salim

et al. 2018

Linear Algebra and its Applications

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Let (A, B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sumis a pair of nonsingular matrices and (A 1 , B 1 ), . . . , (A t , B t ) are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of (A, B) under congruence over an algebraically closed field of characteristic not 2.

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“…Often symmetric matrix pencils appear as a result of symmetric linearizations for symmetric matrix polynomials [1,23]. Many of these applications require computing eigenstructures of matrix pencils, for example, via a structured staircase form for symmetric matrix pencils [6] as well as understanding the behaviour of these eigenstructures under low rank [30] and general perturbations, and that is where our miniversal deformations may be useful [7,9,20]. Moreover, based on the versal deformation theory, a constructive approach to determine the geometry of the singularities (orientation in space, magnitudes of angles, etc.)…”

Section: Introductionmentioning

confidence: 99%

“…In particular, miniversal deformations of symmetric matrix pencils can help us to construct their stratifications, i.e. closure hierarchies of orbits and bundles, see the examples in [7,9,20]. These stratifications are illustrated by the graphs showing all canonical forms that the symmetric matrix pencils may have in arbitrarily small neighbourhoods of a given symmetric matrix pencil.…”

Section: Introductionmentioning

confidence: 99%

Miniversal deformations of pairs of symmetric matrices under congruence

Dmytryshyn

2019

Linear Algebra and its Applications

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For each pair of complex symmetric matrices (A, B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices (Ã,B), close to (A, B) can be reduced by congruence transformation that smoothly depends on the entries ofÃ andB. Such a normal form is called a miniversal deformation of (A, B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A, B) to its miniversal deformation.

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Miniversal deformations of pairs of skew-symmetric matrices under congruence

Cited by 11 publications

References 29 publications

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

Miniversal deformations of pairs of symmetric matrices under congruence

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