On Hua's fundamental theorem of the geometry of rectangular matrices (Q1604394)

Two \(m\times n\)-matrices \(A\) and \(B\) over a field \(F\) are called adjacent if, and only if, \(\text{rank} A-B=1\). A well-known result due to \textit{L. K. Hua} [Trans. Am. Math. Soc. 61, 229-255 (1947; Zbl 0037.39205)] says that a bijection \(\varphi:F^{m\times n}\to F^{m\times n}\) (\(m,n\geq 2\)) which preserves adjacency in both directions is of the form \(X\mapsto P\cdot f(X)\cdot Q+R\) or, but only when \(m=n\), of the form \(X\mapsto P\cdot f(X)^T\cdot Q+R\) , where \(f\) is an automorphism of \(F\), \(P\in\text{GL}(m,F)\), \(Q\in\text{GL}(n,F)\), and \(R\in F^{m\times n}\). The author aims at weakening the original assumptions of Hua. On the one hand, it is shown that a mapping \(\varphi:F^{m\times n}\to F^{m\times n}\) that preserves adjacency in both directions is injective. On the other hand, it is shown that for real matrices the assumptions in Hua's theorem can be weakened considerably: it is sufficient to assume that \(\varphi:F^{m\times n}\to F^{m\times n}\) is just a mapping with the following property: \(A\) adjacent \(B\) \(\Leftrightarrow\) \(\varphi(A)\) adjacent \(\varphi(B)\). Over the complex numbers a similar result is not possible, since there are non-surjective monomorphisms of the field of complex numbers.