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Orbit closure hierarchies of skew-symmetric matrix pencils
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
2014 (engelsk)Inngår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, nr 4, s. 1429-1443Artikkel i tidsskrift (Fagfellevurdert) Published
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.

sted, utgiver, år, opplag, sider
2014. Vol. 35, nr 4, s. 1429-1443
Emneord [en]
skew-symmetric matrix pencil, stratification, canonical structure information, orbit, bundle
HSV kategori
Identifikatorer
URN: urn:nbn:se:umu:diva-98914DOI: 10.1137/140956841ISI: 000346843200010Scopus ID: 2-s2.0-84919931822OAI: oai:DiVA.org:umu-98914DiVA, id: diva2:784021
Forskningsfinansiär
eSSENCE - An eScience CollaborationSwedish Research Council, A0581501Tilgjengelig fra: 2015-01-28 Laget: 2015-01-28 Sist oppdatert: 2023-03-23bibliografisk kontrollert
Inngår i avhandling
1. Tools for Structured Matrix Computations: Stratifications and Coupled Sylvester Equations
Åpne denne publikasjonen i ny fane eller vindu >>Tools for Structured Matrix Computations: Stratifications and Coupled Sylvester Equations
2015 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Umeå: Umeå universitet, 2015. s. 29
Serie
Report / UMINF, ISSN 0348-0542 ; 15.18
HSV kategori
Identifikatorer
urn:nbn:se:umu:diva-111641 (URN)978-91-7601-379-3 (ISBN)
Disputas
2015-12-11, MA 121 MIT-building, Umeå universitet, Umeå, 13:00 (engelsk)
Opponent
Veileder
Forskningsfinansiär
Swedish Research Council, E0485301Swedish Research Council, A0581501eSSENCE - An eScience Collaboration
Tilgjengelig fra: 2015-11-20 Laget: 2015-11-18 Sist oppdatert: 2018-06-07bibliografisk kontrollert

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