Additive Maps Preserving Rank Equivalence On Hermitian Matrix Spaces

Matrix preserver problems mainly study the maps preserving some invariants fromone matrix space V1 to another matrix space V2. The invariants include functions,subsets, relation, transformation, etc. As this problem has wide practical applicationbackgrounds in other areas, for example, di?erential equation, system control, quantummechanics and mathematical statistics, it becomes one of the active subjects in matrixtheory.Hermitian matrix is an important class of matrices in linear algebra, the study ofpreserving problems on Hermitian matrix space is valuable. In this thesis,we study themaps preserving rank equivalence on Hermitian matrix space Hn(D) over division ringD. The main results obtained in this thesis are as follows:1.We introduce the origin and the development of preserving problems first, andthen introduce the present development of preserving rank equivalence on matrix space.2.We characterize the additive surjective map preserving rank equivalence onHn(D).3.We give some applications of preserving rank equivalence on Hn(D).