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  • 1.
    Belitskii, Genrich
    et al.
    Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel.
    Dmytryshyn, Andrii
    Faculty of Mechanics and Mathematics, Kiev National Taras Shevchenko University, Kiev, Ukraine.
    Lipyanski, Ruvim
    Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel.
    Sergeichuk, Vladimir
    Institute of Mathematics, Kiev, Ukraine.
    Tsurkov, Arkady
    Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel.
    Problems of classifying associative or Lie algebras over a field of characteristic not 2 and finite metabelian groups are wild2009Ingår i: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 18, s. 516-529, artikel-id 41Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let F be a field of characteristic different from 2. It is shown that the problems of classifying

    (i) local commutative associative algebras over F with zero cube radical,

    (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and

    (iii) finite p-groups of exponent p with central commutator subgroup of order  are hopeless since each of them contains

    • the problem of classifying symmetric bilinear mappings UxU → V , or

    • the problem of classifying skew-symmetric bilinear mappings UxU → V ,

    in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.

  • 2.
    Dmytryshyn, Andrii
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Miniversal deformations of pairs of symmetric matrices under congruence2019Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 568, s. 84-105Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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 ((A) over tilde (B) over tilde), close to (A, B) can be reduced by congruence transformation that smoothly depends on the entries of (A ) over tilde and (B) over tilde. 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.

  • 3.
    Dmytryshyn, Andrii
    Örebro universitet, Institutionen för naturvetenskap och teknik. Department of Computing Science, Umeå University, Umeå, Sweden.
    Skew-symmetric matrix pencils: stratification theory and tools2014Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    Investigating the properties, explaining, and predicting the behaviour of a physical system described by a system (matrix) pencil often require the understanding of how canonical structure information of the system pencil may change, e.g., how eigenvalues coalesce or split apart, due to perturbations in the matrix pencil elements. Often these system pencils have different block-partitioning and / or symmetries. We study changes of the congruence canonical form of a complex skew-symmetric matrix pencil under small perturbations. The problem of computing the congruence canonical form is known to be ill-posed: both the canonical form and the reduction transformation depend discontinuously on the entries of a pencil. Thus it is important to know the canonical forms of all such pencils that are close to the investigated pencil. One way to investigate this problem is to construct the stratification of orbits and bundles of the pencils. To be precise, for any problem dimension we construct the closure hierarchy graph for congruence orbits or bundles. Each node (vertex) of the graph represents an orbit (or a bundle) and each edge represents the cover/closure relation. Such a relation means that there is a path from one node to another node if and only if a skew-symmetric matrix pencil corresponding to the first node can be transformed by an arbitrarily small perturbation to a skew-symmetric matrix pencil corresponding to the second node. From the graph it is straightforward to identify more degenerate and more generic nearby canonical structures. A necessary (but not sufficient) condition for one orbit being in the closure of another is that the first orbit has larger codimension than the second one. Therefore we compute the codimensions of the congruence orbits (or bundles). It is done via the solutions of an associated homogeneous system of matrix equations. The complete stratification is done by proving the relation between equivalence and congruence for the skew-symmetric matrix pencils. This relation allows us to use the known result about the stratifications of general matrix pencils (under strict equivalence) in order to stratify skew-symmetric matrix pencils under congruence. Matlab functions to work with skew-symmetric matrix pencils and a number of other types of symmetries for matrices and matrix pencils are developed and included in the Matrix Canonical Structure (MCS) Toolbox.

  • 4.
    Dmytryshyn, Andrii
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Structure preserving stratification of skew-symmetric matrix polynomials2015Rapport (Övrigt vetenskapligt)
    Abstract [en]

    We study how elementary divisors and minimal indices of a skew-symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients. We investigate these changes qualitatively by constructing the stratifications (closure hierarchy graphs) of orbits and bundles for skew-symmetric linearizations. We also derive the necessary and sufficient conditions for the existence of a skew-symmetric matrix polynomial with prescribed degree, elementary divisors, and minimal indices.

  • 5.
    Dmytryshyn, Andrii
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Structure preserving stratification of skew-symmetric matrix polynomials2017Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 532, s. 266-286Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study how elementary divisors and minimal indices of a skew-symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients. We investigate these changes qualitatively by constructing the stratifications (closure hierarchy graphs) of orbits and bundles for skew-symmetric linearizations. We also derive the necessary and sufficient conditions for the existence of a skew-symmetric matrix polynomial with prescribed degree, elementary divisors, and minimal indices.

  • 6.
    Dmytryshyn, Andrii
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Tools for Structured Matrix Computations: Stratifications and Coupled Sylvester Equations2015Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    Developing theory, algorithms, and software tools for analyzing matrix pencils whose matrices have various structures are contemporary research problems. Such matrices are often coming from discretizations of systems of differential-algebraic equations. Therefore preserving the structures in the simulations as well as during the analyses of the mathematical models typically means respecting their physical meanings and may be crucial for the applications. This leads to a fast development of structure-preserving methods in numerical linear algebra along with a growing demand for new theories and tools for the analysis of structured matrix pencils, and in particular, an exploration of their behaviour under perturbations. In many cases, the dynamics and characteristics of the underlying physical system are defined by the canonical structure information, i.e. eigenvalues, their multiplicities and Jordan blocks, as well as left and right minimal indices of the associated matrix pencil. Computing canonical structure information is, nevertheless, an ill-posed problem in the sense that small perturbations in the matrices may drastically change the computed information. One approach to investigate such problems is to use the stratification theory for structured matrix pencils. The development of the theory includes constructing stratification (closure hierarchy) graphs of orbits (and bundles) that provide qualitative information for a deeper understanding of how the characteristics of underlying physical systems can change under small perturbations. In turn, for a given system the stratification graphs provide the possibility to identify more degenerate and more generic nearby systems that may lead to a better system design.

    We develop the stratification theory for Fiedler linearizations of general matrix polynomials, skew-symmetric matrix pencils and matrix polynomial linearizations, and system pencils associated with generalized state-space systems. The novel contributions also include theory and software for computing codimensions, various versal deformations, properties of matrix pencils and matrix polynomials, and general solutions of matrix equations. In particular, the need of solving matrix equations motivated the investigation of the existence of a solution, advancing into a general result on consistency of systems of coupled Sylvester-type matrix equations and blockdiagonalizations of the associated matrices.

  • 7.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Dopico, Froilán M.
    Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain.
    Generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree2017Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 535, s. 213-230Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The set POLd,rm×n of m×n complex matrix polynomials of grade d and (normal) rank at most r in a complex (d+1)mn dimensional space is studied. For r=1,...,min{m,n}−1, we show that POLd,rm×n is the union of the closures of the rd+1 sets of matrix polynomials with rank r, degree exactly d, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. r=min{m,n} and mn, we show that POLd,rm×n coincides with the closure of a single set of the polynomials with rank r, degree exactly d, and the described complete eigenstructure. These complete eigenstructures correspond to generic m×n matrix polynomials of grade d and rank at most r.

  • 8.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Dopico, Froilán M.
    Departamento de Matemáticas, Universidad Carlos III de Madrid.
    Generic matrix polynomials with fixed rank and fixed degree2016Rapport (Övrigt vetenskapligt)
  • 9.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Dopico, Froilán M.
    Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain.
    Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade2018Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 536, s. 1-18Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that the set of m×m complex skew-symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of odd grade d and rank at most 2r. In particular, this result includes the case of skew-symmetric matrix pencils (d=1).

  • 10.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Fonseca, Carlos
    Department of Mathematics, Kuwait University, Kuwait City, Kuwait.
    Rybalkina, Tetiana
    Institute of Mathematics, Kiev, Ukraine.
    Classification of pairs of linear mappings between two vector spaces and between their quotient space and subspace2016Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 509, s. 228-246Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We classify pairs of linear mappings (U -> V, U/U' -> V') in which U, V are finite dimensional vector spaces over a field IF, and U', are their subspaces. (C) 2016 Elsevier Inc. All rights reserved.

  • 11.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Futorny, Vyacheslav
    Department of Mathematics, University of São Paulo, São Paulo, Brazil.
    Klymchuk, Tetiana
    Universitat Politècnica de Catalunya, Barcelona, Spain; Taras Shevchenko National University, Kiev, Ukraine.
    Sergeichuk, Vladimir V.
    Institute of Mathematics, Kiev, Ukraine.
    Generalization of Roth's solvability criteria to systems of matrix equations2017Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 527, s. 294-302Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    W.E. Roth (1952) proved that the matrix equation AX - XB = C has a solution if and only if the matrices [Graphics] and [Graphics] are similar. A. Dmytryshyn and B. Kagstrom (2015) extended Roth's criterion to systems of matrix equations A(i)X(i')M(i) - (NiXi"Bi)-B-sigma i = Ci (i = 1,..., s) with unknown matrices X1,, X-t, in which every X-sigma is X, X-T, or X*. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations. (C) 2017 Elsevier Inc. All rights reserved.

  • 12.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Futorny, Vyacheslav
    Department of Mathematics, University of São Paulo, São Paulo, Brazil.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Klimenko, Lena
    National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kiev, Ukraine.
    Sergeichuk, Vladimir
    Institute of Mathematics, Kiev, Ukraine.
    Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence2015Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 469, s. 305-334Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We construct the Hasse diagrams G2 and G3 for the closure ordering on the sets of congruence classes of 2 × 2 and 3 × 3 complex matrices. In other words, we construct two directed graphs whose vertices are 2 × 2 or, respectively, 3 × 3 canonical matrices under congruence, and there is a directed path from A to B if and only if A can be transformed by an arbitrarily small perturbation to a matrix that is congruent to B. A bundle of matrices under congruence is defined as a set of square matrices A for which the pencils A + λAT belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of 2 × 2 or 3 × 3 matrices have the same properties with respect to perturbations. We construct the Hasse diagrams G2 B and G3 B for the closure ordering on the sets of congruence bundles of 2 × 2 and, respectively, 3 × 3 matrices. We find the isometry groups of 2 × 2 and 3 × 3 congruence canonical matrices.

  • 13.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Futorny, Vyacheslav
    Department of Mathematics, University of São Paulo, São Paulo, Brazil.
    Sergeichuk, Vladimir
    Institute of Mathematics, Kiev, Ukraine.
    Miniversal deformations of matrices of bilinear forms2012Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 436, nr 7, s. 2670-2700Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct miniversal deformations of matrices under congruence.

  • 14.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Futorny, Vyacheslav
    Department of Mathematics, University of São Paulo, São Paulo, Brazil.
    Sergeichuk, Vladimir
    Institute of Mathematics, Kiev, Ukraine.
    Miniversal deformations of matrices under *congruence and reducing transformations2014Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 446, nr April, s. 388-420Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Arnold (1971) [1] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We give miniversal deformations of matrices of sesquilinear forms; that is, of square complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by Dmytryshyn, Futorny, and Sergeichuk (2012) [11].

  • 15.
    Dmytryshyn, Andrii
    et al.
    Dept. Computing Science, Umeå University, Umeå, Sweden.
    Johansson, Stefan
    Dept. Computing Science, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Dept. Computing Science, Umeå University, Umeå, Sweden.
    Canonical structure transitions of system pencils2015Rapport (Övrigt vetenskapligt)
    Abstract [en]

    We investigate the changes under small perturbations of the canonical structure information for a system pencil (A B C D) − s (E 0 0 0), det(E) ≠ 0, associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformation. The results allow to track possible changes under small perturbations of important linear system characteristics.

  • 16.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Johansson, Stefan
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Canonical structure transitions of system pencils2017Ingår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 38, nr 4, s. 1249-1267Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We investigate the changes of the canonical structure information under small perturbations for a system pencil associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformations. The results allow us to track possible changes of important linear system characteristics under small perturbations.

  • 17.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå. University, Umeå, Sweden.
    Johansson, Stefan
    Department of Computing Science and HPC2N, Umeå. University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå. University, Umeå, Sweden.
    Codimension computations of congruence orbits of matrices, symmetric and skew-symmetric matrix pencils using Matlab2013Rapport (Övrigt vetenskapligt)
    Abstract [en]

    Matlab functions to work with the canonical structures for congru-ence and *congruence of matrices, and for congruence of symmetricand skew-symmetric matrix pencils are presented. A user can providethe canonical structure objects or create (random) matrix examplesetups with a desired canonical information, and compute the codi-mensions of the corresponding orbits: if the structural information(the canonical form) of a matrix or a matrix pencil is known it isused for the codimension computations, otherwise they are computednumerically. Some auxiliary functions are provided too. All thesefunctions extend the Matrix Canonical Structure Toolbox.

  • 18.
    Dmytryshyn, Andrii
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Department of Computing Science, Umeå University, Umeå, Sweden.
    Johansson, Stefan
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Van Dooren, Paul
    Department of Mathematical Engineering, Université catholique de Louvain, Louvain-la-Neuve, Belgium.
    Geometry of Matrix Polynomial Spaces2019Ingår i: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials’ orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality.

  • 19.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Johansson, Stefan
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Van Dooren, Paul
    Universite catholique de Louvain, Belgium.
    Geometry of spaces for matrix polynomial Fiedler linearizations2015Rapport (Övrigt vetenskapligt)
    Abstract [en]

    We study how small perturbations of matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy graphs (stratifications) of orbits and bundles of matrix polynomial Fiedler linearizations. We show that the stratifica-tion graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler lineariza-tions have the same geometry (topology). The results are illustrated by examples using the software tool StratiGraph.

  • 20.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Coupled Sylvester-type Matrix Equations and Block Diagonalization2015Ingår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 36, nr 2, s. 580-593Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and $\star$-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of $2 \times 2$ block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations.

  • 21.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Orbit closure hierarchies of skew-symmetric matrix pencils2014Rapport (Övrigt vetenskapligt)
    Abstract [en]

    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. This 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.

  • 22.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Orbit closure hierarchies of skew-symmetric matrix pencils2014Ingår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, nr 4, s. 1429-1443Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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 - lambda B can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C - lambda D if and only if A - lambda B can be approximated by pencils congruent to C - lambda D.

  • 23.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Sergeichuk, Vladimir V.
    Institute of Mathematics, Kiev, Ukraine.
    Skew-symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations2013Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, nr 8, s. 3375-3396Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The homogeneous system of matrix equations (X(T)A + AX, (XB)-B-T + BX) = (0, 0), where (A, B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution and calculate the codimension of the orbit of (A, B) under congruence. These results will be useful in the development of the stratification theory for orbits of skew-symmetric matrix pencils.

  • 24.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Sergeichuk, Vladimir V.
    Ukrainian Acad Sci, Kiev, Ukraine.
    Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations2014Ingår i: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 27, s. 1-18Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.

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