Study Of Some Problems Of Geometry Of Rectangular Matrices

The study of the geometry of matrices was initiated by Hua L.-K. in the middleforties of the last century. From then on, mathematicians at home and abroad ob-tained many good results on the conditions reduction and equivalent conditions forthe geometry of rectangular matrices. In 2004, Huang W.-L. and Wan Z.-X. provedthat an adjacency preserving map is an adjacency preserving map in both directionson the geometry of rectangular matrices over a division ring. In 2006, Huang L.-P.proved the fundamental theorem of the geometry of rectangular matrices over a Be-zout domain with some restrictions. But there are some problems over a division ringof charecteristic two.Based on these work, this paper researches the equivalence of some maps on thegeometry of rectangular matrices. There are three chapters in this paper. In Chapter1, we introduce the background, the developments of recent researches and the mainresult of the paper. In Chapter 2, we prove some properties on the finite field F₂^m×nand research equivalence of di?erent maps of the geometry of rectangular matrices overthe field F₂, and prove that (F₂^m×n,～) is not a good distance graph. If R is a domainin which every finitely generated left (right) ideal is principal, then R is called a Bezoutdomain. In Chapter 3, we get a su?cient conditions of a good distance graph, andprove the following result:Let R be a Bezout domain such that R = F₂, and let m,n be positive integers≥2. Then the graph G := (R^m×n,～) is a good distance graph, where A～Brank(A-B) = 1 for all A,B∈R^m×n.