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Skew-symmetric matrix pencils: stratification theory and tools
Umeå University, Faculty of Science and Technology, Department of Computing Science.
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
ISBN: 978-91-7601-003-7OAI: oai:DiVA.org:umu-87501DiVA: diva2:709589
##### Supervisors
Available from: 2014-04-02 Created: 2014-04-02 Last updated: 2014-04-02Bibliographically approved
##### List of papers
1. Skew-symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations
Open this publication in new window or tab >>Skew-symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations
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.

Elsevier, 2013
##### Keyword
Pair of skew-symmetric matrices, Matrix equations, Orbits, Codimension
Mathematics
##### Identifiers
urn:nbn:se:umu:diva-68465 (URN)10.1016/j.laa.2012.11.025 (DOI)000316521500015 ()
##### Funder
eSSENCE - An eScience CollaborationSwedish Research Council, A0581501 Available from: 2013-04-25 Created: 2013-04-22 Last updated: 2016-08-25Bibliographically approved
2. Codimension computations of congruence orbits of matrices, symmetric and skew-symmetric matrix pencils using Matlab
Open this publication in new window or tab >>Codimension computations of congruence orbits of matrices, symmetric and skew-symmetric matrix pencils using Matlab
##### 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.

##### Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2013. 41 p.
##### Series
Report / UMINF, ISSN 0348-0542 ; 13.18
##### Keyword
Congruence; *congruence; Symmetric matrix pencils; Skew-symmetric matrix pencils; Orbits; Codimension; MATLAB
##### National Category
Computer Science Computational Mathematics
##### Research subject
Numerical Analysis; Computer Science
##### Identifiers
urn:nbn:se:umu:diva-80524 (URN)
Available from: 2013-09-19 Created: 2013-09-19 Last updated: 2015-11-19Bibliographically approved
3. Orbit closure hierarchies of skew-symmetric matrix pencils
Open this publication in new window or tab >>Orbit closure hierarchies of skew-symmetric matrix pencils
##### 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.

##### Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2014. 18 p.
##### Series
, UMINF, ISSN 0348-0542 ; 14.02
##### Keyword
skew-symmetric matrix pencil, stratification, canonical structure information, orbits
##### National Category
Computer Science Computational Mathematics
##### Identifiers
urn:nbn:se:umu:diva-87500 (URN)
Available from: 2014-04-02 Created: 2014-04-02 Last updated: 2014-04-02Bibliographically approved

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