# The Optimal Version of Hua’s Fundamental Theorem of Geometry of Rectangular Matrices

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*Peter Šemrl*

Hua's fundamental theorem of geometry of matrices describes
the general form of bijective maps on the space of all \(m\times n\)
matrices over a division ring \(\mathbb{D}\) which preserve adjacency in
both directions. Motivated by several applications the author studies a long
standing open problem of possible improvements.

There are three natural questions. Can we replace the assumption
of preserving adjacency in both directions by the weaker assumption of
preserving adjacency in one direction only and still get the same
conclusion? Can we relax the bijectivity assumption? Can we obtain an
analogous result for maps acting between the spaces of rectangular
matrices of different sizes?

A division ring is said to be EAS if it is not isomorphic to any
proper subring. For matrices over EAS division rings the author solves all
three problems simultaneously, thus obtaining the optimal version of
Hua's theorem. In the case of general division rings he gets such an
optimal result only for square matrices and gives examples showing that
it cannot be extended to the non-square case.