Automorphisms of \(M_{n}\), partially ordered by rank subtractivity ordering
Let \(M_n\) be the space of all \(n{\times}n\) matrices with coefficients in \(\mathbb{R}\) or \(\mathbb{C}\), where \(n\geqslant 3\). The rank subtractivity or minus partial order on \(M_n\) is defined by \(A\leqslant_{{\text{rs}}}B\) iff \({\text{rank}}\,B = {\text{rank}}\,A+{\text{rank}}\,(B-A)\). This order, restricted to idempotents, induces the usual order on idempotents, i.e. \(P={P^2}_{{\text{rs}}}\,Q=Q^2\) iff \(PQ=QP=P\). The author characterizes bijective mappings \(\Phi\) on \(M_n\) such that \(A\leqslant_{{\text{rs}}}B\) iff \(\Phi(A)\leqslant_{{\text{rs}}}\Phi(B), \; (A,B \in M_n)\).
scientific article; zbMATH DE number 2117288
scientific article