Linearizations of rational matrices from general representations

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