Block minimal bases ℓ-ifications of matrix polynomials

Martínez Dopico, Froilán César; Pérez Álvaro, Javier; Van Dooren, Paul M.

Publication:
Block minimal bases ℓ-ifications of matrix polynomials

dc.affiliation.dpto	UC3M. Departamento de Matemáticas	es
dc.affiliation.grupoinv	UC3M. Grupo de Investigación: Matemática Aplicada a Control, Sistemas y Señales	es
dc.contributor.author	Martínez Dopico, Froilán César
dc.contributor.author	Pérez Álvaro, Javier
dc.contributor.author	Van Dooren, Paul M.
dc.contributor.funder	Ministerio de Economía y Competitividad (España)	es
dc.date.accessioned	2021-04-06T09:10:11Z
dc.date.available	2021-04-06T09:10:11Z
dc.date.issued	2019-02-01
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.	en
dc.format.extent	42
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.	en
dc.identifier.doi	https://doi.org/10.1016/j.laa.2018.10.010
dc.identifier.issn	0024-3795
dc.identifier.publicationfirstpage	163
dc.identifier.publicationlastpage	204
dc.identifier.publicationtitle	Linear Algebra and Its Applications	en
dc.identifier.publicationvolume	562
dc.identifier.uri	https://hdl.handle.net/10016/32268
dc.identifier.uxxi	AR/0000022354
dc.language.iso	eng	en
dc.publisher	Elsevier	en
dc.relation.projectID	Gobierno de España. MTM2015-65798-P	es
dc.relation.projectID	Gobierno de España. MTM2015-68805-REDT	es
dc.relation.projectID	Gobierno de España. MTM2017-90682-REDT	es
dc.rights	© 2018 Elsevier Inc.	en
dc.rights	Atribución-NoComercial-SinDerivadas 3.0 España	*
dc.rights.accessRights	open access	en
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/3.0/es/	*
dc.subject.eciencia	Matemáticas	es
dc.subject.other	Matrix polynomial	en
dc.subject.other	Minimal indices	en
dc.subject.other	Dual minimal bases	en
dc.subject.other	Linearization	en
dc.subject.other	Quadratification	en
dc.subject.other	Dual minimal bases matrix polynomial	en
dc.subject.other	Block kronecker matrix polynomial	en
dc.subject.other	Strong ℓ-ification	en
dc.subject.other	Companion ℓ-ification	en
dc.title	Block minimal bases ℓ-ifications of matrix polynomials	en
dc.type	research article	*
dc.type.hasVersion	AM	*
dspace.entity.type	Publication