Study Of The Geometry Of Symmetric Matrices Over A Commutative Principal Domain

The study of the geometry of matrices was initiated by Hua L.-K. in forties of the 20th century. In 1949, Hua proved the fundamental theorem of the geome-try of symmetric matrices over a field of characteristic not two by the method of constructing involutions. In the mid nineties of the last century, Wan Z.-X. proved the fundamental theorem of the geometry of symmetric matrices over any field by the method of maximal sets. In 2006, Huang L.-P. proved the fundamental theo-rem of the geometry of symmetric matrices over a commutative principal domain under some condition. But the fundamental theorem of the geometry of symmetric matrices over a commutative principal domain under general condition is still an open and difficult problem. In this paper, we discuss the fundamental theorem of the geometry of symmetric matrices over a commutative principal domain.Let R be a commutative principal domain. Denote by Sn(R) the set of all n x n symmetric matrices over R. This paper has three chapters. In Chapter 1, we introduce the background and the status of recent researches. In Chapter 2, we mainly discuss the fundamental theorem of the geometry of symmetric matrices over R which is not Jacobson semisimple. In Chapter 3, we prove the fundamental theorem of the geometry of 2×2 symmetric matrices over R under new conditions, and obtain the main result of this. paper as follow:Let R be a commutative principal domain, and letφ:S2(R)(?) S2(R) be an adjacency preserving bijective map in both directions such thatφ(Mi)= Mi,φ(Li)=Li,i=1,2, where Mi and Ci are the standard maximal sets of rank 1 and rank 2, respectively. Then there exists a fixed invertible matrix P∈GL2(R) such that whereα∈R* is fixed, andσis an automorphism of R.