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Skew-symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
2013 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, no 8, 3375-3396 p.Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2013. Vol. 438, no 8, 3375-3396 p.
Keyword [en]
Pair of skew-symmetric matrices, Matrix equations, Orbits, Codimension
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-68465DOI: 10.1016/j.laa.2012.11.025ISI: 000316521500015OAI: oai:DiVA.org:umu-68465DiVA: diva2:618018
Funder
eSSENCE - An eScience CollaborationSwedish Research Council, A0581501
Available from: 2013-04-25 Created: 2013-04-22 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Skew-symmetric matrix pencils: stratification theory and tools
Open this publication in new window or tab >>Skew-symmetric matrix pencils: stratification theory and tools
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
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.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2014. 12 p.
Series
UMINF, ISSN 0348-0542 ; 14.05
National Category
Computer Science Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-87501 (URN)978-91-7601-003-7 (ISBN)
Supervisors
Available from: 2014-04-02 Created: 2014-04-02 Last updated: 2014-04-02Bibliographically approved
2. Tools for Structured Matrix Computations: Stratifications and Coupled Sylvester Equations
Open this publication in new window or tab >>Tools for Structured Matrix Computations: Stratifications and Coupled Sylvester Equations
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
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.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2015. 29 p.
Series
Report / UMINF, ISSN 0348-0542 ; 15.18
National Category
Computer and Information Science
Identifiers
urn:nbn:se:umu:diva-111641 (URN)978-91-7601-379-3 (ISBN)
Public defence
2015-12-11, MA 121 MIT-building, Umeå universitet, Umeå, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, E0485301Swedish Research Council, A0581501eSSENCE - An eScience Collaboration
Available from: 2015-11-20 Created: 2015-11-18 Last updated: 2015-12-02Bibliographically approved

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Dmytryshyn, AndriiKågström, Bo

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