Root vectors of polynomial and rational matrices: Theory and computation

The notion of root polynomials of a polynomial matrix P(λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R(λ), possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R(λ), without any need to assume that R(λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R(λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ₀. If λ₀ is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.