The Research On Idempotent Preservers In Hermite Matrix Space

Idempotent preserving problem is a very important research area,In 1897,Frobenius first studied the linear preserving problem.With the deepening of this research,many scholars began to study the preserving problems by weakening some conditions.after decades of development,idempotent preserving problem has very rich achievements.Hermite matrix is an important research object.Hermite matrix also has practical applications in quantum mechanics.In recent years,many scholars concern the research from the low dimensional preserving problems to the high dimensional preserving problems.In this paper,we give preserving idempotent relationship mapping in different dimensional hermitian matrix space.Two conditions φ(Ett)=0 and φ(Ett)≠0 are discussed.Through finding basis and finding special idempotent matrix,by mathematical induction,when n<m,gave a description of the maping satisfy that if A-λB is idempotent,we gave the φ(A)-λφ(B)is idempotent from n×n dimensional hermitian matrix space to m×m dimensional matrix space.Finally,the result was generalized to the case of tripotent preserving.Two conditions ψ(Ett)=0 and ψ(Ett)≠0 are discussed.Through finding basis and finding special tripotent matrix,gave a description of the map satisfy that if A-λB is tripotent,the ψ(A)-λψ(B)is tripotent from 2×2 dimensional hermitian matrix space to the m×m dimensional full matrix space is described.In the field of preserving problems,the characterization of different dimensional nonlinear maps of idempotent preserving problems is much more complex than that of different dimensional linear maps.We still have many problems that can be studied.