Citation:
Dopico, F. M. & Noferini, V. (2020). Root polynomials and their role in the theory of matrix polynomials. Dopico, F. M. & Noferini, V. (2020). Root polynomials and their role in the theory of matrix polynomials. Linear Algebra and Its Applications, 584, pp. 37–78.
xmlui.dri2xhtml.METS-1.0.item-contributor-funder:
Ministerio de Economía y Competitividad (España)
Sponsor:
Supported by “Ministerio de Economía, Industria y Competitividad of Spain” and “Fondo Europeo de
Desarrollo Regional (FEDER) of EU” through grants MTM2015-65798-P and MTM2017-90682-REDT
Project:
Gobierno de España. MTM2015-65798-P Gobierno de España. MTM2017-90682-REDT
Keywords:
Matrix polynomials
,
Polynomial matrices
,
Root polynomials
,
Complete set
,
Maximal set
,
Minimaximal set
,
Eigenvalues
,
Minimal bases
,
Singular
,
Eigenvectors of singular matrix polynomials
,
Jordan chains
,
μ-independent set
We develop a complete and rigorous theory of root polynomials of arbitrary matrix polynomials, i.e., either regular or singular, and study how these vector polynomials are related to the spectral properties of matrix polynomials. We pay particular attention toWe develop a complete and rigorous theory of root polynomials of arbitrary matrix polynomials, i.e., either regular or singular, and study how these vector polynomials are related to the spectral properties of matrix polynomials. We pay particular attention to the so-called maximal sets of root polynomials and prove that they carry complete information about the eigenvalues (finite or infinite) of matrix polynomials and that they are related to the matrices that transform any matrix polynomial into its Smith form. In addition, we describe clearly, for the first time in the literature, the extremality properties of such maximal sets and identify some of them whose vectors have minimal grade. Once the main theoretical properties of root polynomials have been established, the interaction of root polynomials with three problems that have attracted considerable attention in the literature is analyzed. More precisely, we study the change of root polynomials under rational transformations, or reparametrizations, of matrix polynomials, the recovery of the root polynomials of a matrix polynomial from those of its most important linearizations, and the relationship between the root polynomials of two dual pencils. We emphasize that for the case of regular matrix polynomials all the results in this paper can be translated into the language of Jordan chains, as a consequence of the well known relationship between root polynomials and Jordan chains. Therefore, a number of open problems are also solved for Jordan chains of regular matrix polynomials. We also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials.[+][-]