The Study Of Geometry Of Rectangular Block Triangular Matrices

The study of the geometry of matrices was in_itiated by L.K.Hua in the mid fortiesof the last century. The triangular matrices play an important role in Lie algebra, thegeometry of triangular matrices is an important content in the geometry of matrices.In 2006, Huang Li-Ping and Cai Yong-Yu proved the fundamental theorem of geometryof block triangular matrices over division ring with different ways, and their results aresimpler than the former study. But it is a open problem that the case of rectangularblock triangular matrices. I continued this work, and solve the fundamental theoremof four rectangular block matrices over a division ring with D≠F₂.The paper mainly introduce three aspects work. 1,We discuss the constructionof maximal sets of on rectangular block triangular matrices over a division ring. 2,Wediscuss some basic properties about the adjacency preserving bijective maps on therectangular block triangular matrix spaces. 3,By the maximal sets and affine geometry,we prove the fundamental theorem of four rectangular block triangular matrices overdivision ring as follows: Let D be a division ring with D≠_F2. Let m₁,n₂≥2be integers. Let D be an adjacency preserving bijective map in both directions onT_(mi,ni,2). If T_(mi,ni,2)≠T₍(n_3-i,m_3-i,2)), thenwhere P∈T_(mi,2),Q∈T_(ni,2) are invertible matrices, T₀∈T_(mi,ni,2),σis an automor-phism of D. When T_(mi,ni,2) = T₍(n_3-i,m3-i,2), in addtion to the former, we also have ananti-automorphismÏ„of D such that...