The Study Of Geometry Of Triangular Matrices

The study of the geometry of matrices was initiated by L. K. Hua in the mid forties of the last century. There are six types for the geometry of matrices studied previously, which were: geometry of the rectangular matrices, symmetric matrices, alternate matrices, Hermitian matrices, skew-Hermitian matrices and triangular matrices. In the geometry of matrices, the points of the associated space are a certain kind of matrices, there is an arithmetic distance of two points in this space, and there is a transformation group acting on this space. Two distinct points is said to be adjacent if their arithmetic distance is one or minimal. The fundamental problem of the geometry of matrices is to characterize the transformation preserving the adjacency of matrix spaces by as few geometric invariants as possible. The geometry of matrices has applications to algebra, geometry and theory of functions of several complex variables. The fundamental problem of the geometry of matrices can be interpreted as a theorem on graph automorphism of the graph on a certain kind of matrices, and it also has practical application in the linear (additive) preserver problems which are the research area in matrix and operator theory.Recently, W.L.Chooi and M.H.Lim have discussed the adjacency preserving mappings on block triangular matrix algebra. In 2006, Li-Ping Huang and Y.Y-Cai proved the fundamental theorem of geometry of block triangular matrices over division ring D with D≠F2 , and their results are simpler than former study,but it is a open problem that the situation over F2 . I continued this work, which can be seen as the fundamental study of the geometry of matrices. The content of this text is composed of three chapters. In the first chapter, we mainly introduce the background and the status of recent researches of the geometry of matrices. In the second chapter, we discussed the construction of maximal sets on block triangular matrices over a division ring. In the third chapter, we used the finite and the fundamental theorem of geometry of rectangular matrices to prove the fundamental theorem of geometry of block triangular matrices over the field F2 .