Recovery of eigenvectors of rational matrix functions from Fiedler-like linearizations

Linearization is a standard method often used when dealing with matrix polynomials. Recently, the concept of linearization has been extended to rational matrix functions and Fiedler-like matrix pencils for rational matrix functions have been constructed. A linearization L(λ)L(λ) of a rational matrix function G(λ)G(λ) does not necessarily guarantee a simple way of recovering eigenvectors of G(λ)G(λ) from those of L(λ)L(λ). We show that Fiedler-like pencils of G(λ)G(λ) allow an easy operation free recovery of eigenvectors of G(λ)G(λ), that is, eigenvectors of G(λ)G(λ) are recovered from eigenvectors of Fiedler-like pencils of G(λ)G(λ) without performing any arithmetic operations. We also consider Fiedler-like pencils of the Rosenbrock system polynomial S(λ)S(λ) associated with an LTI system Σ in state-space form (SSF) and show that the Fiedler-like pencils allow operation free recovery of eigenvectors of S(λ)S(λ). The eigenvectors of S(λ)S(λ) are the invariant zero directions of the LTI system Σ.