Hua's fundamental theorems of the geometry of matrices and related results (Q1863555)

In the 40's L. K. Hua characterized the bijective maps preserving the pairs \((A,B)\) of matrices such that \(rank(A,B)=1\) . In the paper under review the author provides a shorter proof of Hua's result in the case of square matrices and sketches a proof for the case of rectangular matrices. The main tool is a result due to \textit{P. G. Ovchinnikov} [J. Funct. Anal. 115, 184-189 (1993; Zbl 0806.46069)] characterizing the automorphisms of the poset of idempotent matrices. The author improves Ovchinnikov's result and presents a brief survey of some improvements of Hua's theorem as well as of some related open questions on the subject.