Recovery of minimal bases and minimal indices of rational matrices from Fiedler-like pencils.

Abstract Let G (λ) be an n × n rational matrix. By considering a minimal realization of G (λ) , Fiedler-like pencils (such as Fiedler pencils, generalized Fiedler pencils and Fiedler pencils with repetition) of G (λ) have been proposed recently which are shown to be linearizations of G (λ). We show that the Fiedler-like pencils allow operation-free recovery of eigenvectors and minimal bases of G (λ) , that is, eigenvectors and minimal bases of G (λ) can be recovered from those of the Fiedler-like pencils of G (λ) without performing any arithmetic operations. Further, we show that the minimal indices of G (λ) can be easily recovered from those of the Fiedler-like pencils of G (λ). We also show that a Fiedler pencil with repetition of G (λ) can be constructed directly from that of a matrix polynomial. [ABSTRACT FROM AUTHOR]