Hua's fundamental theorem of the geometry of matrices.
Let \(M_n(F)\) be the space of all \(n\times n\) matrices over the field \(F\), \(n\geq 2\). Two matrices, \(A, B\in M_n(F)\) are adjacent if \(\text{rank}(A-B)= 1\). \textit{L. K. Hua}'s fundamental theorem of the geometry of square matrices [A theorem on matrices over a field and its applications, Acta Math. Sinica 1, 109--163 (1951)] characterizes bijective maps on \(M_n(F)\) that preserve adjacency in both directions. In this paper the author treats an open problem whether the result of Hua holds true under the weaker assumption of preserving the adjacency in one direction only. The author answers this question in the affirmative in the case that every nonzero homomorphism \(f:F\to F\) is surjective. In order to prove this result the author has to improve \textit{P. G. Ovchinnikov}'s characterization of automorphisms of the poset of idempotent matrices [J. Funct. Anal. 115, No. 1, 184--189 (1993; Zbl 0806.46069)]. The author uses a method to improve the infinite-dimensional part of Ovchinnikov's result and characterizes injective continuous order-preserving maps on complex idempotent matrices. The author concludes the paper by describing some applications of the results.
scientific article; zbMATH DE number 2055654
scientific article