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Tools for Structured Matrix Computations: Stratifications and Coupled Sylvester Equations
Umeå University, Faculty of Science and Technology, Department of Computing Science.
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
ISBN: 978-91-7601-379-3 (print)OAI: oai:DiVA.org:umu-111641DiVA: diva2:872408
##### Public defence
2015-12-11, MA 121 MIT-building, Umeå universitet, Umeå, 13:00 (English)
##### 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
##### List of papers
1. Coupled Sylvester-type Matrix Equations and Block Diagonalization
Open this publication in new window or tab >>Coupled Sylvester-type Matrix Equations and Block Diagonalization
2015 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 36, no 2, 580-593 p.Article in journal (Refereed) 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.

##### Keyword
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
##### National Category
Computer Science Mathematical Analysis
##### Identifiers
urn:nbn:se:umu:diva-107104 (URN)10.1137/151005907 (DOI)000357407800011 ()
Available from: 2015-09-23 Created: 2015-08-18 Last updated: 2015-11-19Bibliographically approved
2. 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
3. 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
4. Orbit closure hierarchies of skew-symmetric matrix pencils
Open this publication in new window or tab >>Orbit closure hierarchies of skew-symmetric matrix pencils
2014 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, no 4, 1429-1443 p.Article in journal (Refereed) 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.

##### Keyword
skew-symmetric matrix pencil, stratification, canonical structure information, orbit, bundle
Computer Science
##### Identifiers
urn:nbn:se:umu:diva-98914 (URN)10.1137/140956841 (DOI)000346843200010 ()
##### Funder
eSSENCE - An eScience CollaborationSwedish Research Council, A0581501 Available from: 2015-01-28 Created: 2015-01-28 Last updated: 2015-11-19Bibliographically approved
5. Geometry of spaces for matrix polynomial Fiedler linearizations
Open this publication in new window or tab >>Geometry of spaces for matrix polynomial Fiedler linearizations
##### 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.

28 p.
##### Series
Report / UMINF, ISSN 0348-0542 ; 15.17
##### National Category
Mathematics Computer and Information Science
##### Identifiers
urn:nbn:se:umu:diva-111639 (URN)
##### Funder
Swedish Research Council, E0485301eSSENCE - An eScience Collaboration Available from: 2015-11-18 Created: 2015-11-18 Last updated: 2015-11-19Bibliographically approved
6. Structure preserving stratification of skew-symmetric matrix polynomials
Open this publication in new window or tab >>Structure preserving stratification of skew-symmetric matrix polynomials
##### 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.

##### Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2015. 26 p.
##### Series
Report / UMINF, ISSN 0348-0542 ; 15.16
##### National Category
Natural Sciences Mathematics Computer and Information Science
##### Identifiers
urn:nbn:se:umu:diva-111634 (URN)
##### Funder
Swedish Research Council, E0485301eSSENCE - An eScience Collaboration Available from: 2015-11-18 Created: 2015-11-18 Last updated: 2015-11-19Bibliographically approved
7. Canonical structure transitions of system pencils
Open this publication in new window or tab >>Canonical structure transitions of system pencils
##### 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.

26 p.
##### Series
Report / UMINF, ISSN 0348-0542 ; 15.15
##### Keyword
linear system, descriptor system, state-space system, system pencil, matrix pencil, orbit, bundle, perturbation, versal deformation, stratification
##### National Category
Mathematics Computer and Information Science Electrical Engineering, Electronic Engineering, Information Engineering Civil Engineering
##### Identifiers
urn:nbn:se:umu:diva-111632 (URN)
##### Funder
eSSENCE - An eScience CollaborationSwedish Research Council, E048530 Available from: 2015-11-18 Created: 2015-11-18 Last updated: 2015-11-19Bibliographically approved

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Cite
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v. 2.23.1
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