Linear transformations between matrix spaces that map one rank specific set into another Original Research Article

We characterize, in several instances, those linear transformations from the linear space of m×n matrices into the linear space of p×q matrices that map the set of matrices having a fixed rank into the set of matrices having a fixed rank. Examples are given showing that, in contrast with the case of linear transformations on the linear space of m×n matrices mapping a rank specific set into itself, in the more general case of linear transformations between two full matrix spaces, often one cannot expect neat and predictable results.