Linearizations of Rational Matrices from General Representations
Linear algebra and its applications(2022)
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.
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Key words
Rational matrix,Rational eigenvalue problem,Block minimal bases pencil,Linearization in a set,Linearization at infinity,Grade
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