Linear Maps Preserving Rank Equivalence On The Module Of Hermitian Matrices Over PID

As one of the active research subjects in matrix theory, the study of preserverproblems，especially the linear preserver problems, can be traced back to more thanone hundred years ago. After the development of so many years, we have achievedgreat success and established a relatively complete theoretical system.Hermitian matrix is a type of important matrix. Thus, the study of preserverproblems on Hermitian matrix over rings is also very interesting. In this thesis, wedo research on the linear maps which preserving rank equivalence on the module ofHermitian matrix H n(R)over principal ideal domain(PID), and get the followingconclusions:1. We introduce the backgrounds and the history of preserving problems firstly.Then we describe the current research situation of the preservers of rank equivalenceon matrix spaces.2. We study the linear maps which preserving rank one on H n(R), the module ofthe Hermitian matrix over PID.3. We discuss the linear maps which preserving rank equivalence on H n(R).