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Coupled Sylvester-type Matrix Equations and Block Diagonalization
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå Univ, HPC2N, SE-90187 Umeå, Sweden.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå Univ, HPC2N, SE-90187 Umeå, Sweden.
2015 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 36, no 2, p. 580-593Article 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.

##### Place, publisher, year, edition, pages
2015. Vol. 36, no 2, p. 580-593
##### Keywords [en]
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 Sciences Mathematical Analysis
##### Identifiers
ISI: 000357407800011OAI: oai:DiVA.org:umu-107104DiVA, id: diva2:856028
Available from: 2015-09-23 Created: 2015-08-18 Last updated: 2018-06-07Bibliographically approved
##### In thesis
1. 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. p. 29
##### Series
Report / UMINF, ISSN 0348-0542 ; 15.18
##### National Category
Computer and Information Sciences
##### 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)
##### Funder
Swedish Research Council, E0485301Swedish Research Council, A0581501eSSENCE - An eScience Collaboration Available from: 2015-11-20 Created: 2015-11-18 Last updated: 2018-06-07Bibliographically approved

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Dmytryshyn, AndriiKågstrom, Bo

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SIAM Journal on Matrix Analysis and Applications
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