A short solution to the subpencil problem involving only column minimal indices

We give an explicit numerical characterization in terms of classical Kronecker invariants of the subpencil relation between two matrix pencils determined only by minimal indices for columns (respectively for rows). In addition, a method to construct the completion pencils is also presented (thus solving the pencil completion problem completely in this special case). We showcase our method by carrying out the computations in a well-explained example. Theoretically, our criteria could also be obtained as a particular case of the very technical and involved results by Dodig and Stošić (from [6] and [5]). However, our self-contained approach is easy to follow and independent of the technical results used in the mentioned papers.