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Strongly minimal self-conjugate linearizations for polynomial and rational matrices

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dc.affiliation.dptoUC3M. Departamento de Matemáticas
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Matemática Aplicada a Control, Sistemas y Señaleses
dc.contributor.authorMartínez Dopico, Froilán César
dc.contributor.authorQuintana Ponce, María del Carmen
dc.contributor.authorVan Dooren, Paul
dc.contributor.funderComunidad de Madrides
dc.contributor.funderMinisterio de Economía y Competitividad (España)es
dc.contributor.funderMinisterio de Ciencia e Innovación (España)es
dc.contributor.funderUniversidad Carlos III de Madrides
dc.date.accessioned2023-02-23T13:40:40Z
dc.date.available2023-02-23T13:40:40Z
dc.date.issued2022-09
dc.description.abstractWe prove that we can always construct strongly minimal linearizations of an arbitrary rational matrix from its Laurent expansion around the point at infinity, which happens to be the case for polynomial matrices expressed in the monomial basis. If the rational matrix has a particular self-conjugate structure, we show how to construct strongly minimal linearizations that preserve it. The structures that are considered are the Hermitian and skew-Hermitian rational matrices with respect to the real line, and the para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary axis. We pay special attention to the construction of strongly minimal linearizations for the particular case of structured polynomial matrices. The proposed constructions lead to efficient numerical algorithms for constructing strongly minimal linearizations. The fact that they are valid for any rational matrix is an improvement on any other previous approach for constructing other classes of structure preserving linearizations, which are not valid for any structured rational or polynomial matrix. The use of the recent concept of strongly minimal linearization is the key for getting such generality. Strongly minimal linearizations are Rosenbrock's polynomial system matrices of the given rational matrix, but with a quadruple of linear polynomial matrices (i.e., pencils): L(λ):=[A(λ)C(λ)−B(λ)D(λ)], where A(λ) is regular, and the pencils [A(λ)−B(λ)] and [A(λ)C(λ)] have no finite or infinite eigenvalues. Strongly minimal linearizations contain the complete information about the zeros, poles, and minimal indices of the rational matrix and allow one to very easily recover its eigenvectors and minimal bases. Thus, they can be combined with algorithms for the generalized eigenvalue problem for computing the complete spectral information of the rational matrix.en
dc.description.sponsorshipThis work is part of the "Proyecto de I+D+i PID2019-106362GB-I00 financiado por MCIN/AEI/10.13039/501100011033." This work has been also funded by "Ministerio de Economía,Industria y Competitividad (MINECO)" of Spain through grant MTM2017-90682-REDT and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), in the V PRICIT (Regional Programme of Research and Technological Innovation). The second author was funded by the "contrato predoctoral" BES-2016-076744 of MINECO and by an Academy of Finland grant (Suomen Akatemian päätös 331240). This work was partially developed while the third author held a "Chairof Excellence UC3M - Banco de Santander" at Universidad Carlos III de Madrid in the academic year 2019-2020.en
dc.format.extent28
dc.identifier.bibliographicCitationDopico, F. M., Quintana, M. C., & Dooren, P. V. (2022b). Strongly Minimal Self-Conjugate Linearizations for Polynomial and Rational Matrices. SIAM Journal on Matrix Analysis and Applications, 43(3), 1354-1381.en
dc.identifier.doihttps://doi.org/10.1137/21M1453542
dc.identifier.issn0895-4798
dc.identifier.publicationfirstpage1354
dc.identifier.publicationissue3
dc.identifier.publicationlastpage1381
dc.identifier.publicationtitleSIAM Journal on Matrix Analysis and Applicationsen
dc.identifier.publicationvolume43
dc.identifier.urihttps://hdl.handle.net/10016/36656
dc.identifier.uxxiAR/0000032129
dc.language.isoeng
dc.publisherSIAMen
dc.relation.projectIDGobierno de España. BES-2016-076744es
dc.relation.projectIDGobierno de España. PID2019-106362GB-I00es
dc.relation.projectIDGobierno de España. MTM2017-90682-REDTes
dc.relation.projectIDComunidad de Madrid. EPUC3M23es
dc.rights© 2022, Society for Industrial and Applied Mathematicsen
dc.rights.accessRightsopen accessen
dc.subject.ecienciaMatemáticases
dc.subject.otherStructured realizationsen
dc.subject.otherStructured linearizationsen
dc.subject.otherStrong minimalityen
dc.subject.otherSelf-conjugaterational matricesen
dc.titleStrongly minimal self-conjugate linearizations for polynomial and rational matricesen
dc.typeresearch article*
dc.type.hasVersionVoR*
dspace.entity.typePublication
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