Linear Preserver Problems On Tensor Product Spaces Of Symmetric Matrices

In the field of basic mathematics,the study of invariants plays an important role.Preserver problem are to characterize maps which preserve certain invariants on a given structure.When the given structure is matrix,characterizing maps of preserving invariants between matrix sets is called matrix preserver problem.Matrix preserver problem has a wide range of practical applications in quantum mechanics,differential geometry,differential equations,system control and mathematical statistics and other fields.Many scholars classify the preserver problems from the aspects of variables,matrix sets,maps properties and so on.A large number of literature emerge,and the research results tend to be perfect.In 2012,Professor Li Chi-Kwong presented linear preserver problems on tensor product spaces of matrices at the International Conference on Matrix and Operator Theory,which provided a new research direction for preserver problems.The thesis focuses on the study of linear preserver problems on tensor product spaces of symmetric matrices.Firstly,a set of basis of symmetric matrix are found.Secondly,the images of the tensor product of the basis and any symmetric matrix are characterized.Finally,since any matrix can be written as a linear combination of basis,the image of tensor product of any two symmetric matrices can be characterized.Thus,the form of linear map which preserves rank of tensor product of symmetric matrices in complex field and the form of linear map which preserves idempotent of tensor product of symmetric matrices in real field are characterized.As an application,the form of linear map which preserves tripotent of tensor products of symmetric matrices is also characterized.The thesis obtains the conclusions that two important problems(rank preserver and idempotence preserver)are added to the tensor product limit.It is shown that the linear maps of preserving rank and idempotence in symmetric matrix tensor product space are the same as that of classically preserving rank and idempotence,that is,the same map is characterized by fewer matrices.The conclusions also enrich the existing theories of linear preserver problems on the matrix tensor product space.