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  • 1.
    Dmytryshyn, Andrii
    Umeå universitet, Institutionen för datavetenskap.
    Miniversal deformations of pairs of skew-symmetric matrices under congruence2016In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 506, p. 506-534Article in journal (Refereed)
    Abstract [en]

    Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair (A, B) we provide a normal form with a minimal number of independent parameters to which all pairs of skew-symmetric matrices ((A) over tilde (,) (B) over tilde), close to (A, B) can be reduced by congruence transformation which smoothly depends on the entries of the matrices in the pair ((A) over tilde (,) (B) over tilde). An upper bound on the distance from such a miniversal deformation to (A, B) is derived too. We also present an example of using miniversal deformations for analyzing changes in the canonical structure information (i.e. eigenvalues and minimal indices) of skew-symmetric matrix pairs under perturbations.

  • 2.
    Dmytryshyn, Andrii
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Miniversal deformations of pairs of symmetric matrices under congruence2019In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 568, p. 84-105Article in journal (Refereed)
    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
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Structure preserving stratification of skew-symmetric matrix polynomials2017In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 532, p. 266-286Article in journal (Refereed)
    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.

  • 4.
    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 degree2017In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 535, p. 213-230Article in journal (Refereed)
    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.

  • 5.
    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 grade2018In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 536, p. 1-18Article in journal (Refereed)
    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).

  • 6.
    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 subspace2016In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 509, p. 228-246Article in journal (Refereed)
    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.

  • 7.
    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 equations2017In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 527, p. 294-302Article in journal (Refereed)
    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.

  • 8.
    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 congruence2015In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 469, p. 305-334Article in journal (Refereed)
    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.

  • 9.
    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 forms2012In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 436, no 7, p. 2670-2700Article in journal (Refereed)
    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.

  • 10.
    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 transformations2014In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 446, no April, p. 388-420Article in journal (Refereed)
    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].

  • 11.
    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 equations2013In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, no 8, p. 3375-3396Article in journal (Refereed)
    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.

  • 12.
    Pereira, Rajesh
    et al.
    Department of Mathematics and Statistics, University of Guelph, Guelph ON, Canada.
    Rush, Stephen
    Department of Mathematics and Statistics, University of Guelph, Guelph ON, Canada.
    Wielandt's theorem, spectral sets, and Banach algebras2013In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 439, no 4, p. 852-855Article in journal (Refereed)
    Abstract [en]

    Let A be a complex unital Banach algebra and let a,b∈A. We give regions of the complex plane which contain the spectrum of a+b or ab using von Neumann spectral set theory. These results are a direct generalization of a theorem of Wielandt on the eigenvalues of the sum of two normal matrices.

1 - 12 of 12
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