Block minimal bases ℓ-ifications of matrix polynomials

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Show simple item record Martínez Dopico, Froilán César Pérez Álvaro, Javier Van Dooren, Paul M. 2021-04-06T09:10:11Z 2021-04-06T09:10:11Z 2019-02-01
dc.identifier.bibliographicCitation Dopico, F. M., Pérez, J. & Van Dooren, P. (2019). Block minimal bases ℓ-ifications of matrix polynomials. Linear Algebra and Its Applications, 562, pp. 163–204.
dc.identifier.issn 0024-3795
dc.description.abstract The standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil, known as a strong linearization. This process transforms the problem into an equivalent generalized eigenvalue problem. However, there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials. This has motivated the idea of a strong ℓ-ification of a matrix polynomial, which is a matrix polynomial of degree at most ℓ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial. We present in this work a novel method for constructing strong ℓ-ifications of matrix polynomials of size m x n and grade d when ℓ < d, and ℓ divides nd or md. This method is based on a family called "strong block minimal bases matrix polynomials", and relies heavily on properties of dual minimal bases. We show how strong block minimal bases ℓ-ifications can be constructed from the coefficients of a given matrix polynomial P(lambda). We also show that these t-ifications satisfy many desirable properties for numerical applications: they are strong ℓ-ifications regardless of whether P(lambda) is regular or singular, the minimal indices of the ℓ-ifications are related to those of P(lambda) via constant uniform shifts, and eigenvectors and minimal bases of P(lambda) can be recovered from those of any of the strong block minimal bases tifications. In the special case where ℓ divides d, we introduce a subfamily of strong block minimal bases matrix polynomials named "block Kronecker matrix polynomials", which is shown to be a fruitful source of companion ℓ-ifications.
dc.format.extent 42
dc.language.iso eng
dc.publisher Elsevier
dc.rights © 2018 Elsevier Inc.
dc.rights Atribución-NoComercial-SinDerivadas 3.0 España
dc.subject.other Matrix polynomial
dc.subject.other Minimal indices
dc.subject.other Dual minimal bases
dc.subject.other Linearization
dc.subject.other Quadratification
dc.subject.other Dual minimal bases matrix polynomial
dc.subject.other Block kronecker matrix polynomial
dc.subject.other Strong ℓ-ification
dc.subject.other Companion ℓ-ification
dc.title Block minimal bases ℓ-ifications of matrix polynomials
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.type.version acceptedVersion
dc.identifier.publicationfirstpage 163
dc.identifier.publicationlastpage 204
dc.identifier.publicationtitle Linear Algebra and Its Applications
dc.identifier.publicationvolume 562
dc.identifier.uxxi AR/0000022354
dc.contributor.funder Ministerio de Economía y Competitividad (España)
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