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  • 1.
    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.

  • 2.
    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 pencils2015Report (Other academic)
    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.

  • 3.
    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 pencils2017In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 38, no 4, p. 1249-1267Article in journal (Refereed)
    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.

  • 4.
    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 Matlab2013Report (Other academic)
    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.

  • 5.
    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 linearizations2015Report (Other academic)
    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.

  • 6.
    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 Diagonalization2015In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 36, no 2, p. 580-593Article in journal (Refereed)
    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.

  • 7.
    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 pencils2014Report (Other academic)
    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.

  • 8.
    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 pencils2014In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, no 4, p. 1429-1443Article in journal (Refereed)
    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.

  • 9.
    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.

  • 10.
    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 equations2014In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 27, p. 1-18Article in journal (Refereed)
    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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