Linear transformations between matrix spaces that map one rank specific set into another (Q1855415)

Let \(F\) be a field. The problem is how to characterize the linear transformations \(\varphi:M_{m\times n}(F)\to M_{p\times q}(F)\) which map the set of matrices having a fixed rank into the set of matrices having a fixed rank. Examples show that this problem is probably intractable in full generality. In this paper, the authors characterizes all the linear transformations \(\varphi\) which map any matrix \(A\in M_{m\times n}(F)\) with rank \(A=1\) to a matrix with rank \(\varphi(A)=1\). Furthermore, the characterization of linear transformations \(\varphi\) having the property that rank \(A=k\iff \operatorname {rank}\varphi(A)=k\) under certain conditions is obtained.