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1. Linearizations of rational matrices from general representations.

Author

Pérez, Javier and Quintana, María C.

Subjects
*MATRICES (Mathematics), *POLYNOMIALS, *EIGENVALUES, *EIGENVECTORS
Abstract

We construct a new family of linearizations of rational matrices R (λ) written in the general form R (λ) = D (λ) + C (λ) A (λ) − 1 B (λ) , where D (λ) , C (λ) , B (λ) and A (λ) are polynomial matrices. Such representation always exists and is not unique. The new linearizations are constructed from linearizations of the polynomial matrices D (λ) and A (λ) , where each of them can be represented in terms of any polynomial basis. In addition, we show how to recover eigenvectors, when R (λ) is regular, and minimal bases and minimal indices, when R (λ) is singular, from those of their linearizations in this family. [ABSTRACT FROM AUTHOR]

Published
2022
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2. Linearizations of rational matrices from general representations

Author

Javier Pérez, María C. Quintana, University of Montana, Department of Mathematics and Systems Analysis, Aalto-yliopisto, and Aalto University

Subjects
Numerical Analysis, Algebra and Number Theory, Recovery of minimal bases, Grade, Linearization at infinity, Recovery of eigenvectors, Numerical Analysis (math.NA), Block minimal bases pencil, 65F15, 15A18, 15A22, 15A54, 93B18, 93B60, FOS: Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematics - Numerical Analysis, Rational eigenvalue problem, Linearization in a set, Rational matrix, Recovery of minimal indices
Abstract

Funding Information: Supported by ?Ministerio de Econom?a, Industria y Competitividad (MINECO)? of Spain and ?Fondo Europeo de Desarrollo Regional (FEDER)? of EU through grants MTM2015-65798-P and MTM2017-90682-REDT, and the predoctoral contract BES-2016-076744 of MINECO. Supported by an Academy of Finland grant (Suomen Akatemian p??t?s 331230). Publisher Copyright: © 2022 Elsevier Inc. We construct a new family of linearizations of rational matrices R(λ) written in the general form R(λ)=D(λ)+C(λ)A(λ)−1B(λ), where D(λ), C(λ), B(λ) and A(λ) are polynomial matrices. Such representation always exists and is not unique. The new linearizations are constructed from linearizations of the polynomial matrices D(λ) and A(λ), where each of them can be represented in terms of any polynomial basis. In addition, we show how to recover eigenvectors, when R(λ) is regular, and minimal bases and minimal indices, when R(λ) is singular, from those of their linearizations in this family.

Published
2020

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