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Tools for Structured Matrix Computations: Stratifications and Coupled Sylvester Equations
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
2015 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Umeå: Umeå universitet , 2015. , s. 29
Serie
Report / UMINF, ISSN 0348-0542 ; 15.18
Nationell ämneskategori
Data- och informationsvetenskap
Identifikatorer
URN: urn:nbn:se:umu:diva-111641ISBN: 978-91-7601-379-3 (tryckt)OAI: oai:DiVA.org:umu-111641DiVA, id: diva2:872408
Disputation
2015-12-11, MA 121 MIT-building, Umeå universitet, Umeå, 13:00 (Engelska)
Opponent
Handledare
Forskningsfinansiär
Vetenskapsrådet, E0485301Vetenskapsrådet, A0581501eSSENCE - An eScience CollaborationTillgänglig från: 2015-11-20 Skapad: 2015-11-18 Senast uppdaterad: 2018-06-07Bibliografiskt granskad
Delarbeten
1. Coupled Sylvester-type Matrix Equations and Block Diagonalization
Öppna denna publikation i ny flik eller fönster >>Coupled Sylvester-type Matrix Equations and Block Diagonalization
2015 (Engelska)Ingår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 36, nr 2, s. 580-593Artikel i tidskrift (Refereegranskat) Published
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.

Nyckelord
matrix equation, Sylvester equation, Stein equation, Roth's theorem, nsistency, block diagonalization, MMEL JW, 1987, LINEAR ALGEBRA AND ITS APPLICATIONS, V88-9, P139 anat R., 2007, BIT NUMERICAL MATHEMATICS, V47, P763
Nationell ämneskategori
Datavetenskap (datalogi) Matematisk analys
Identifikatorer
urn:nbn:se:umu:diva-107104 (URN)10.1137/151005907 (DOI)000357407800011 ()
Tillgänglig från: 2015-09-23 Skapad: 2015-08-18 Senast uppdaterad: 2018-06-07Bibliografiskt granskad
2. Skew-symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations
Öppna denna publikation i ny flik eller fönster >>Skew-symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations
2013 (Engelska)Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, nr 8, s. 3375-3396Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
Elsevier, 2013
Nyckelord
Pair of skew-symmetric matrices, Matrix equations, Orbits, Codimension
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:umu:diva-68465 (URN)10.1016/j.laa.2012.11.025 (DOI)000316521500015 ()
Externt samarbete:
Forskningsfinansiär
eSSENCE - An eScience CollaborationVetenskapsrådet, A0581501
Tillgänglig från: 2013-04-25 Skapad: 2013-04-22 Senast uppdaterad: 2018-06-08Bibliografiskt granskad
3. Codimension computations of congruence orbits of matrices, symmetric and skew-symmetric matrix pencils using Matlab
Öppna denna publikation i ny flik eller fönster >>Codimension computations of congruence orbits of matrices, symmetric and skew-symmetric matrix pencils using Matlab
2013 (Engelska)Rapport (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Umeå: Umeå Universitet, 2013. s. 41
Serie
Report / UMINF, ISSN 0348-0542 ; 13.18
Nyckelord
Congruence; *congruence; Symmetric matrix pencils; Skew-symmetric matrix pencils; Orbits; Codimension; MATLAB
Nationell ämneskategori
Datavetenskap (datalogi) Beräkningsmatematik
Forskningsämne
numerisk analys; datalogi
Identifikatorer
urn:nbn:se:umu:diva-80524 (URN)
Tillgänglig från: 2013-09-19 Skapad: 2013-09-19 Senast uppdaterad: 2018-06-08Bibliografiskt granskad
4. Orbit closure hierarchies of skew-symmetric matrix pencils
Öppna denna publikation i ny flik eller fönster >>Orbit closure hierarchies of skew-symmetric matrix pencils
2014 (Engelska)Ingår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, nr 4, s. 1429-1443Artikel i tidskrift (Refereegranskat) 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.

Nyckelord
skew-symmetric matrix pencil, stratification, canonical structure information, orbit, bundle
Nationell ämneskategori
Datavetenskap (datalogi)
Identifikatorer
urn:nbn:se:umu:diva-98914 (URN)10.1137/140956841 (DOI)000346843200010 ()
Forskningsfinansiär
eSSENCE - An eScience CollaborationVetenskapsrådet, A0581501
Tillgänglig från: 2015-01-28 Skapad: 2015-01-28 Senast uppdaterad: 2018-06-07Bibliografiskt granskad
5. Geometry of spaces for matrix polynomial Fiedler linearizations
Öppna denna publikation i ny flik eller fönster >>Geometry of spaces for matrix polynomial Fiedler linearizations
2015 (Engelska)Rapport (Övrigt vetenskapligt)
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.

Förlag
s. 28
Serie
Report / UMINF, ISSN 0348-0542 ; 15.17
Nationell ämneskategori
Matematik Data- och informationsvetenskap
Identifikatorer
urn:nbn:se:umu:diva-111639 (URN)
Forskningsfinansiär
Vetenskapsrådet, E0485301eSSENCE - An eScience Collaboration
Tillgänglig från: 2015-11-18 Skapad: 2015-11-18 Senast uppdaterad: 2018-06-07Bibliografiskt granskad
6. Structure preserving stratification of skew-symmetric matrix polynomials
Öppna denna publikation i ny flik eller fönster >>Structure preserving stratification of skew-symmetric matrix polynomials
2015 (Engelska)Rapport (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Umeå: Umeå universitet, 2015. s. 26
Serie
Report / UMINF, ISSN 0348-0542 ; 15.16
Nationell ämneskategori
Naturvetenskap Matematik Data- och informationsvetenskap
Identifikatorer
urn:nbn:se:umu:diva-111634 (URN)
Forskningsfinansiär
Vetenskapsrådet, E0485301eSSENCE - An eScience Collaboration
Tillgänglig från: 2015-11-18 Skapad: 2015-11-18 Senast uppdaterad: 2018-06-07Bibliografiskt granskad
7. Canonical structure transitions of system pencils
Öppna denna publikation i ny flik eller fönster >>Canonical structure transitions of system pencils
2015 (Engelska)Rapport (Övrigt vetenskapligt)
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.

Förlag
s. 26
Serie
Report / UMINF, ISSN 0348-0542 ; 15.15
Nyckelord
linear system, descriptor system, state-space system, system pencil, matrix pencil, orbit, bundle, perturbation, versal deformation, stratification
Nationell ämneskategori
Matematik Data- och informationsvetenskap Elektroteknik och elektronik Samhällsbyggnadsteknik
Identifikatorer
urn:nbn:se:umu:diva-111632 (URN)
Forskningsfinansiär
eSSENCE - An eScience CollaborationVetenskapsrådet, E048530
Tillgänglig från: 2015-11-18 Skapad: 2015-11-18 Senast uppdaterad: 2018-06-07Bibliografiskt granskad

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