Linear maps preserving I-idempotent matrix on matrix spaces

Suppose F is a field of characteristic not 2.Let Mn(F) be the space of all n×n full matrices over F.A matrix A∈Mn(F) is called Ⅰ-idempotent matrix,if there exists λ∈F and an idempotent matrix M∈Mn(F) such that A=λI+M.For a linear map φ:Mn(F)→Mn(F) and an I-idempotent matrix A,if φ(A) is an I-idempotent matrix,then φ preserves I-idempotent matrices.It is shown that if φ preserves I-idempotent matrices,then for every A∈Mn(F),there exists an invertible matrix P∈Mn(F) and λ∈F such that φ(A)=PAP-1+f(A)I,or φ(A)=PAtP-1+f(A)I.