Affine spaces of strong linearizations for rational matrices and the recovery of eigenvectors and minimal bases

The main aim of this paper is two-fold. First, to define and analyze two affine spaces of matrix pencils associated with a realization of an n×n rational matrix G(λ) and show that almost all of these pencils are strong linearizations of G(λ). Second, to describe the recovery of eigenvectors, minimal bases and minimal indices of G(λ) from those of the strong linearizations of G(λ). The affine spaces of pencils are constructed from the vector spaces of linearizations of matrix polynomials defined and analyzed by Mackey et al. in [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971–1004]. We show that eigenvectors, minimal bases and minimal indices of G(λ) can be easily recovered from those of the strong linearizations of G(λ). In particular, we construct a symmetric strong linearization of G(λ) when G(λ) is regular and symmetric. We also describe the recovery of pole-zero structure of G(λ) at infinity from the pole-zero structure at infinity of a strong linearization of G(λ).