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Dmytryshyn, A., Johansson, S., Kågström, B. & Van Dooren, P. (2019). Geometry of Matrix Polynomial Spaces. Foundations of Computational Mathematics
Open this publication in new window or tab >>Geometry of Matrix Polynomial Spaces
2019 (English)In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383Article in journal (Refereed) Epub ahead of print
Abstract [en]

We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials’ orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality.

Place, publisher, year, edition, pages
Springer-Verlag New York, 2019
Keywords
Matrix polynomials Stratifications Matrix pencils, Fiedler linearization, Canonical structure information, Orbit, Bundle
National Category
Mathematics Computational Mathematics Computer and Information Sciences
Identifiers
urn:nbn:se:oru:diva-74859 (URN)10.1007/s10208-019-09423-1 (DOI)
Projects
VR E0485301eSSENCE
Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-08-19Bibliographically approved
Dmytryshyn, A. (2019). Miniversal deformations of pairs of symmetric matrices under congruence. Linear Algebra and its Applications, 568, 84-105
Open this publication in new window or tab >>Miniversal deformations of pairs of symmetric matrices under congruence
2019 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 568, p. 84-105Article in journal (Refereed) Published
Abstract [en]

For each pair of complex symmetric matrices (A, B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices ((A) over tilde (B) over tilde), close to (A, B) can be reduced by congruence transformation that smoothly depends on the entries of (A ) over tilde and (B) over tilde. Such a normal form is called a miniversal deformation of (A, B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A, B) to its miniversal deformation.

Place, publisher, year, edition, pages
Elsevier, 2019
Keywords
Symmetric matrix pair, Symmetric matrix pencil, Congruence canonical form, Perturbation, Versal formation, Codimension
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:oru:diva-74875 (URN)10.1016/j.laa.2018.05.034 (DOI)000462111400005 ()2-s2.0-85048551989 (Scopus ID)
Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-09-20Bibliographically approved
Dmytryshyn, A. & Dopico, F. M. (2018). Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade. Linear Algebra and its Applications, 536, 1-18
Open this publication in new window or tab >>Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade
2018 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 536, p. 1-18Article in journal (Refereed) Published
Abstract [en]

We show that the set of m×m complex skew-symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of odd grade d and rank at most 2r. In particular, this result includes the case of skew-symmetric matrix pencils (d=1).

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Complete eigenstructure, Genericity, Matrix polynomials, Skew-symmetry, Normal rank, Orbits, Pencils
National Category
Algebra and Logic
Research subject
business data processing
Identifiers
urn:nbn:se:oru:diva-74880 (URN)10.1016/j.laa.2017.09.006 (DOI)000414814500001 ()2-s2.0-85029546242 (Scopus ID)
Funder
Swedish Research Council, E0485301eSSENCE - An eScience Collaboration
Note

Research funders:

Ministerio de Economía, Industria y Competitividad of Spain

Fondo Europeo de Desarrollo Regional (FEDER) of EU

Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-09-20Bibliographically approved
Dmytryshyn, A., Johansson, S. & Kågström, B. (2017). Canonical structure transitions of system pencils. SIAM Journal on Matrix Analysis and Applications, 38(4), 1249-1267
Open this publication in new window or tab >>Canonical structure transitions of system pencils
2017 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 38, no 4, p. 1249-1267Article in journal (Refereed) Published
Abstract [en]

We investigate the changes of the canonical structure information under small perturbations for a system pencil associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformations. The results allow us to track possible changes of important linear system characteristics under small perturbations.

Keywords
linear system, generalized state-space system, system pencil, matrix pencil, orbit, bundle, perturbation, versal deformation, stratification
National Category
Computer Sciences Mathematical Analysis
Research subject
business data processing
Identifiers
urn:nbn:se:oru:diva-74889 (URN)10.1137/16M1097857 (DOI)000418665600009 ()2-s2.0-85022337450 (Scopus ID)
Funder
Swedish Research Council, E0485301eSSENCE - An eScience Collaboration
Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-09-20Bibliographically approved
Dmytryshyn, A., Futorny, V., Klymchuk, T. & Sergeichuk, V. V. (2017). Generalization of Roth's solvability criteria to systems of matrix equations. Linear Algebra and its Applications, 527, 294-302
Open this publication in new window or tab >>Generalization of Roth's solvability criteria to systems of matrix equations
2017 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 527, p. 294-302Article in journal (Refereed) Published
Abstract [en]

W.E. Roth (1952) proved that the matrix equation AX - XB = C has a solution if and only if the matrices [Graphics] and [Graphics] are similar. A. Dmytryshyn and B. Kagstrom (2015) extended Roth's criterion to systems of matrix equations A(i)X(i')M(i) - (NiXi"Bi)-B-sigma i = Ci (i = 1,..., s) with unknown matrices X1,, X-t, in which every X-sigma is X, X-T, or X*. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations. (C) 2017 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Systems of matrix equations, Sylvester equations, Roth's criteria
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:oru:diva-74884 (URN)10.1016/j.laa.2017.04.011 (DOI)000402344000014 ()2-s2.0-85017553149 (Scopus ID)
Funder
Swedish Research Council, E0485301eSSENCE - An eScience Collaboration
Note

Research funders:

National Council for Scientific and Technological Development CNPq

São Paulo Research Foundation FAPESP

Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-09-20Bibliographically approved
Dmytryshyn, A. & Dopico, F. M. (2017). Generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree. Linear Algebra and its Applications, 535, 213-230
Open this publication in new window or tab >>Generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree
2017 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 535, p. 213-230Article in journal (Refereed) Published
Abstract [en]

The set POLd,rm×n of m×n complex matrix polynomials of grade d and (normal) rank at most r in a complex (d+1)mn dimensional space is studied. For r=1,...,min{m,n}−1, we show that POLd,rm×n is the union of the closures of the rd+1 sets of matrix polynomials with rank r, degree exactly d, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. r=min{m,n} and mn, we show that POLd,rm×n coincides with the closure of a single set of the polynomials with rank r, degree exactly d, and the described complete eigenstructure. These complete eigenstructures correspond to generic m×n matrix polynomials of grade d and rank at most r.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Complete eigenstructure, Genericity, Matrix polynomials, Normal rank, Orbits
National Category
Computer Sciences Algebra and Logic
Identifiers
urn:nbn:se:oru:diva-74881 (URN)10.1016/j.laa.2017.09.007 (DOI)000413058000012 ()2-s2.0-85029308799 (Scopus ID)
Funder
Swedish Research Council, E0485301eSSENCE - An eScience CollaborationStiftelsen Längmanska kulturfonden, BA17-1175
Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-09-20Bibliographically approved
Dmytryshyn, A. (2017). Structure preserving stratification of skew-symmetric matrix polynomials. Linear Algebra and its Applications, 532, 266-286
Open this publication in new window or tab >>Structure preserving stratification of skew-symmetric matrix polynomials
2017 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 532, p. 266-286Article in journal (Refereed) Published
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
Elsevier, 2017
Keywords
Skew-symmetric matrix polynomials, Matrix polynomials, Stratifications, Skew-symmetric matrix pencils, Orbit, Bundle
National Category
Algebra and Logic Computational Mathematics
Research subject
Computer Science; business data processing
Identifiers
urn:nbn:se:oru:diva-74878 (URN)10.1016/j.laa.2017.06.044 (DOI)000411297500017 ()2-s2.0-85022339632 (Scopus ID)
Funder
Swedish Research Council, E0485301eSSENCE - An eScience Collaboration
Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-09-20Bibliographically approved
Dmytryshyn, A., Fonseca, C. & Rybalkina, T. (2016). Classification of pairs of linear mappings between two vector spaces and between their quotient space and subspace. Linear Algebra and its Applications, 509, 228-246
Open this publication in new window or tab >>Classification of pairs of linear mappings between two vector spaces and between their quotient space and subspace
2016 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 509, p. 228-246Article in journal (Refereed) Published
Abstract [en]

We classify pairs of linear mappings (U -> V, U/U' -> V') in which U, V are finite dimensional vector spaces over a field IF, and U', are their subspaces. (C) 2016 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
Elsevier, 2016
Keywords
Canonical forms; Pairs of linear mappings; Matrix pencils
National Category
Computer Sciences
Identifiers
urn:nbn:se:oru:diva-74883 (URN)10.1016/j.laa.2016.07.016 (DOI)000385338000011 ()2-s2.0-84981294489 (Scopus ID)
Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-09-20Bibliographically approved
Dmytryshyn, A. & Dopico, F. M. (2016). Generic matrix polynomials with fixed rank and fixed degree. Umeå: Umeå Universitet
Open this publication in new window or tab >>Generic matrix polynomials with fixed rank and fixed degree
2016 (English)Report (Other academic)
Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2016. p. 18
Series
UMINF, ISSN 0348-0542 ; 16/19
Keywords
Complete eigenstructure, genericity, matrix polynomials, normal rank, orbits
National Category
Mathematics
Identifiers
urn:nbn:se:oru:diva-74882 (URN)
Funder
Swedish Research Council, E0485301eSSENCE - An eScience Collaboration
Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-08-06Bibliographically approved
Dmytryshyn, A., Johansson, S. & Kågström, B. (2015). Canonical structure transitions of system pencils. Umeå: Umeå universitet
Open this publication in new window or tab >>Canonical structure transitions of system pencils
2015 (English)Report (Other academic)
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.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2015. p. 26
Series
UMINF, ISSN 0348-0542 ; 5
Keywords
Linear system, descriptor system, state-space system, system pencil, matrix pencil, orbit, bundle, perturbation, versal deformation, stratification
National Category
Mathematics Computer and Information Sciences Electrical Engineering, Electronic Engineering, Information Engineering Civil Engineering
Identifiers
urn:nbn:se:oru:diva-74888 (URN)
Funder
eSSENCE - An eScience CollaborationSwedish Research Council, E048530
Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-08-06Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-9110-6182

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