Root vectors of polynomial and rational matrices: Theory and computation

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