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We construct a new family of strong linearizations of rational matricesconsidering the polynomial part of them expressed in a basis that satisfies athree term recurrence relation. For this purpose, we combine the theorydeveloped by Amparan et al., MIMS EPrint 2016.51, and the new linearizations ofpolynomial matrices introduced by Fa{\ss}bender and Saltenberger, LinearAlgebra Appl., 525 (2017). In addition, we present a detailed study of how torecover eigenvectors of a rational matrix from those of its linearizations inthis family. We complete the paper by discussing how to extend the results whenthe polynomial part is expressed in other bases, and by presenting stronglinearizations that preserve the structure of symmetric or Hermitian rationalmatrices. A conclusion of this work is that the combination of the results inthis paper with those in Amparan et al., MIMS EPrint 2016.51, allows us to useessentially all the strong linearizations of polynomial matrices developed inthe last fifteen years to construct strong linearizations of any rationalmatrix by expressing such matrix in terms of its polynomial and strictly properparts.

Strong linearizations of rational matrices with polynomial part expressed in an orthogonal basis