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Van Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Data

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Matemática Aplicada a Control, Sistemas y Señaleses
dc.contributor.authorAnguas Márquez, Luis Miguel
dc.contributor.authorMartínez Dopico, Froilán César
dc.contributor.authorHollister, Richard
dc.contributor.authorMackey, Don Steven
dc.contributor.funderMinisterio de Economía y Competitividad (España)es
dc.date.accessioned2021-04-07T07:44:54Z
dc.date.available2021-04-07T07:44:54Z
dc.date.issued2019-06-13
dc.description.abstractThe structural data of any rational matrix R(\lambda ), i.e., the structural indices of its poles and zeros together with the minimal indices of its left and right nonespaces, is known to satisfy a simple condition involving certain sums of these indices. This fundamental constraint was first proved by Van Dooren in 1978; here we refer to this result as the rational index sum theorem. An analogous result for polynomial matrices has been independently discovered (and rediscovered) several times in the past three decades. In this paper we clarify the connection between these two seemingly different index sum theorems, describe a little bit of the history of their development, and discuss their curious apparent unawareness of each other. Finally, we use the connection between these results to solve a fundamental inverse problem for rational matrices---for which lists \scrL of prescribed structural data does there exist some rational matrix R(\lambda ) that realizes exactly the list \scrL ? We show that Van Dooren's condition is the only constraint on rational realizability; that is, a list \scrL is the structural data of some rational matrix R(\lambda ) if and only if \scrL satisfies the rational index sum condition.en
dc.format.extent19
dc.identifier.bibliographicCitationAnguas, L. M., Dopico, F. M., Hollister, R. & Mackey, D. S. (2019). Van Dooren’s Index Sum Theorem and Rational Matrices with Prescribed Structural Data. SIAM Journal on Matrix Analysis and Applications, 40(2), pp. 720–738.en
dc.identifier.doihttps://doi.org/10.1137/18M1171370
dc.identifier.issn0895-4798
dc.identifier.publicationfirstpage720
dc.identifier.publicationissue2
dc.identifier.publicationlastpage738
dc.identifier.publicationtitleSIAM Journal on Matrix Analysis and Applicationsen
dc.identifier.publicationvolume40
dc.identifier.urihttps://hdl.handle.net/10016/32282
dc.identifier.uxxiAR/0000023960
dc.language.isoengen
dc.publisherSociety for Industrial and Applied Mathematics (SIAM)en
dc.relation.projectIDGobierno de España. MTM2015-65798-Pes
dc.relation.projectIDGobierno de España. MTM2015-68805-REDTes
dc.relation.projectIDGobierno de España. MTM2017-90682-REDTes
dc.relation.projectIDGobierno de España. BES-2013-065688es
dc.relation.projectIDGobierno de España. EEBB-I-2016-11462es
dc.rights© 2019, Society for Industrial and Applied Mathematicsen
dc.rights.accessRightsopen accessen
dc.subject.ecienciaMatemáticases
dc.subject.otherEigenvaluesen
dc.subject.otherIndex sum theoremen
dc.subject.otherStructural indicesen
dc.subject.otherRational matricesen
dc.subject.otherPolesen
dc.subject.otherZerosen
dc.subject.otherInvariant ordersen
dc.subject.otherMinimal indicesen
dc.subject.otherPolynomial matricesen
dc.titleVan Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Dataen
dc.typeresearch article*
dc.type.hasVersionAM*
dspace.entity.typePublication
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