Inverse problems of polynomial and rational matrices involving the eigenstructure and data of the column and row spaces

Abstract The eigenstructure of a rational matrix is comprised by its invariant rational functions, both finite and at infinity, and by the minimal indices of its left and right null spaces. These quantities arise in many applications and have been thoroughly studied in numerous references. There are two other fundamental subspaces of rational matrices that, in contrast, have received much less attention: the column and row spaces. They also have their associated minimal indices, which play a key role in the problem of factorizing polynomial matrices into low‐degree factors. This work solves the problems of finding necessary and sufficient conditions for the existence of rational matrices in two scenarios: when the invariant rational functions and the minimal bases or minimal indices of the column and row spaces are prescribed, and when the eigenstructure together with the minimal indices of the column and row spaces are prescribed. The particular, but extremely important, cases of these problems for polynomial matrices are solved first and are the main tool for solving the general problems. The results in this work complete and nontrivially extend the necessary and sufficient conditions recently obtained for the existence of polynomial and rational matrices when only the eigenstructure is prescribed.