Export Mapping Of Guaranteed Matrix Relationships

The matrix preserver problems is an important research areas in matrix theory, which has a better theoretical value and practical significance, and many excellent esults have appeared. Suppose that F is a field and n≥3 are integers. Denote by Mn(F) the set of all n x n matrices over F. Let fij(i∈[1,n],j∈[1,n]) be functions on F, where [1,n] stands for the set{1,...,n}. We say that a map f:Mn(F)â†'Mn(F), is induced by fij if f is defined by f:(aij)â†'(fij(aij)),(?)(aij)∈Mn(F). Suppose(?)and(?)stand for Similar relationship and congruent relationship between of matrices respectively. We say that a map f on Mn(F) preserves similarity if 4～B=>f(A)～f(B),VA,B ∈Mn(F), a map f on Mn(F) preserving inverses of matrices means f(A)f(A-1)= In for every invertible A ∈Mn(F). We say that a map f preserves congruent ifA(?)B=>f(A)(?)f(B).Set the adjoint matrix A* is A.We say that a map f preserves adjoint ifA∈Mn(F),andf(A*)=(f(A))*.In this paper, we characterize induced maps preserving similarity, preserving adjoint, pre-serving inverses of matrices and preserving congruent relationship between matrices on Sn(F) or Kn(F), respectively.