Strong linear preservers of rank reverse permutability on triangular matrices (Q819763)

The authors classify linear mappings on upper triangular matrices that satisfy \[ \text{rk}(A_1\dots A_k)=\text{rk}(A_k\dots A_1)\Longleftrightarrow \text{rk}(\Phi(A_1)\dots \Phi(A_k))=\text{rk}(\Phi(A_k)\dots \Phi(A_1)). \] It is also assumed that the underlying field differs from \(\mathbb{Z}_2\). The proof is a reduction to rank-commutativity preservers (i.e., to the case of \(k=2\)), which further reduces to linear mappings on upper-triangular matrices that preserve rank-one.