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

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Show simple item record Anguas Márquez, Luis Miguel Martínez Dopico, Froilán César Hollister, Richard Mackey, Don Steven 2021-04-07T07:44:54Z 2021-04-07T07:44:54Z 2019-06-13
dc.identifier.bibliographicCitation Anguas, 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.
dc.identifier.issn 0895-4798
dc.description.abstract The 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.
dc.format.extent 19
dc.language.iso eng
dc.publisher Society for Industrial and Applied Mathematics (SIAM)
dc.rights © 2019, Society for Industrial and Applied Mathematics
dc.subject.other Eigenvalues
dc.subject.other Index sum theorem
dc.subject.other Structural indices
dc.subject.other Rational matrices
dc.subject.other Poles
dc.subject.other Zeros
dc.subject.other Invariant orders
dc.subject.other Minimal indices
dc.subject.other Polynomial matrices
dc.title Van Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Data
dc.type article
dc.subject.eciencia Matemáticas
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. MTM2015-65798-P
dc.relation.projectID Gobierno de España. MTM2015-68805-REDT
dc.relation.projectID Gobierno de España. MTM2017-90682-REDT
dc.relation.projectID Gobierno de España. BES-2013-065688
dc.relation.projectID Gobierno de España. EEBB-I-2016-11462
dc.type.version acceptedVersion
dc.identifier.publicationfirstpage 720
dc.identifier.publicationissue 2
dc.identifier.publicationlastpage 738
dc.identifier.publicationtitle SIAM Journal on Matrix Analysis and Applications
dc.identifier.publicationvolume 40
dc.identifier.uxxi AR/0000023960
dc.contributor.funder Ministerio de Economía y Competitividad (España)
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