Inversion of regular analytic matrix functions: Local Smith form and subspace duality

This paper analyzes the relation between the local rank-structure of a regular analytic matrix function and the one of its inverse function. The local rank factorization (lrf) of a matrix function is introduced, which characterizes extended canonical systems of root functions and the local Smith form. An interpretation of the local rank factorization in terms of Jordan chains and Jordan pairs is provided. Duality results are shown to hold between the subspaces associated with the lrf of the matrix function and the one of its reduced adjoint.