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

Let G(lambda) be an n x n rational matrix. By considering a minimal realization of G(lambda), Fiedler-like pencils (such as Fiedler pencils, generalized Fiedler pencils and Fiedler pencils with repetition) of G(lambda) have been proposed recently which are shown to be linearizations of G(lambda). We show that the Fiedler-like pencils allow operation-free recovery of eigenvectors and minimal bases of G(lambda), that is, eigenvectors and minimal bases of G(lambda) can be recovered from those of the Fiedler-like pencils of G(lambda) without performing any arithmetic operations. Further, we show that the minimal indices of G(lambda) can be easily recovered from those of the Fiedler-like pencils of G(lambda). We also show that a Fiedler pencil with repetition of G(lambda) can be constructed directly from that of a matrix polynomial. (C) 2018 Elsevier Inc. All rights reserved.