Block full rank linearizations of rational matrices

Abstract

We introduce a new family of linearizations of rational matrices, which we call block full rank linearizations. The theory of block full rank linearizations is useful as it establishes very simple criteria to determine if a pencil is a linearization of a rational matrix in a target set or in the whole underlying field, by using rank conditions. Block full rank linearizations allow us to recover locally information about zeros and poles. To recover the pole-zero information at infinity, we will define the grade of the new block full rank linearizations as linearizations at infinity and the notion of degree of a rational matrix will be used. Moreover, the eigenvectors of a rational matrix associated with its eigenvalues in a target set can be obtained from the eigenvectors of its block full rank linearizations in that set. This new family of linearizations generalizes and includes the structures appearing in most of the linearizations for rational matrices constructed in the literature. As example, we study the structure and properties of the linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, 2021].