Linear Preserver Problems On Symmetric Matrix Modules Over Rings

The problems to characterize the linear operators which preserve certain functions, subsets, relations or transformations invariants between matrix sets are called"Linear Preserver Problems". Linear Preserver Problem is a very active topic in the field of matrix theory, it has wide applications in other areas, such as, differential equations, systems control, etc. In the recent years, the study on the Linear Preserver Problem has made great progress.After introducing the background and development of the Linear Preserver Problem, we study the problem of linear preserving inverses and idempotent on symmetric matrix modules over rings. The main results obtained in this thesis are as follows:1. We characterized linear maps preserving inverses of matrices from symmetric matrix modules onto matrix modules over a principal ideal domain of characteristic not 3 and suppose 2 is a unit.2. We characterized maps preserving idempotent matrix from symmetric matrix modules to matrix modules over a commutative idempotence-diagonalizable ring and 2, 3, 5 are units.