Root polynomials and their role in the theory of matrix polynomials
Root polynomials and their role in the theory of matrix polynomials
复制标题
根多项式及其在矩阵多项式理论中的作用
DOI：
10.1016/j.laa.2019.09.006
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发表时间：
2020
影响因子：
1.1
通讯作者：
V. Noferini
中科院分区：
文献类型：
--
作者：
F. Dopico;V. Noferini
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 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.
