Linearizations of rational matrices from general representations

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.