Linearizations of rational matrices from general representations

Javier Pérez; María C. Quintana

doi:10.1016/j.laa.2022.04.007

Linearizations of rational matrices from general representations

Javier Pérez
, María C. Quintana

Mathematical Sciences

Aalto University

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

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)⁻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.

Original language	English
Pages (from-to)	89-126
Number of pages	38
Journal	Linear Algebra and Its Applications
Volume	647
DOIs	https://doi.org/10.1016/j.laa.2022.04.007
State	Published - Aug 15 2022

Funding

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).

Funders	Funder number
European Commission	MTM2017-90682-REDT, MTM2015-65798-P, BES-2016-076744
Academy of Finland	331230

Keywords

Block minimal bases pencil
Grade
Linearization at infinity
Linearization in a set
Rational eigenvalue problem
Rational matrix
Recovery of eigenvectors
Recovery of minimal bases
Recovery of minimal indices

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10.1016/j.laa.2022.04.007

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title = "Linearizations of rational matrices from general representations",

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)−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.",

keywords = "Block minimal bases pencil, Grade, Linearization at infinity, Linearization in a set, Rational eigenvalue problem, Rational matrix, Recovery of eigenvectors, Recovery of minimal bases, Recovery of minimal indices",

author = "Javier P{\'e}rez and Quintana, \{Mar{\'i}a C.\}",

note = "Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2022",

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day = "15",

doi = "10.1016/j.laa.2022.04.007",

language = "English",

volume = "647",

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journal = "Linear Algebra and Its Applications",

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TY - JOUR

T1 - Linearizations of rational matrices from general representations

AU - Pérez, Javier

AU - Quintana, María C.

PY - 2022/8/15

Y1 - 2022/8/15

N2 - 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.

AB - 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.

KW - Block minimal bases pencil

KW - Grade

KW - Linearization at infinity

KW - Linearization in a set

KW - Rational eigenvalue problem

KW - Rational matrix

KW - Recovery of eigenvectors

KW - Recovery of minimal bases

KW - Recovery of minimal indices

UR - https://www.scopus.com/pages/publications/85129390075

U2 - 10.1016/j.laa.2022.04.007

DO - 10.1016/j.laa.2022.04.007

M3 - Article

AN - SCOPUS:85129390075

SN - 0024-3795

VL - 647

SP - 89

EP - 126

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Linearizations of rational matrices from general representations

Abstract

Funding

Keywords

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