Linearizations of rational matrices from general representations

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 are 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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