全矩阵空间上保持Ⅰ-幂等矩阵的线性映射

摘要

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 φφ 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.%设F是特征不为2的任意域,Mn(F)表示F上所有n×n矩阵所组成的空间.对任意A∈Mn(F),若存在λ∈F和幂等阵M∈Mn(F)使得A=λI+M,则称A为I-幂等矩阵.设φ:Mn(F)→Mn(F)为线性映射,若当A为I-幂等矩阵时,φ(A)也为I-幂等矩阵,则称φ保持I-幂等矩阵.刻画Mn(F)上保持I-幂等矩阵的线性双射的形式,即若φ:Mn(F)→Mn(F)为保持I-幂等矩阵的线性双射,则对任意A∈Mn(F),存在可逆阵P∈Mn(F)和线性泛函f:Mn(F)→F使得φ(A)=PAP-1+f(A)I或φ(A)=PAtP-1+f(A)I.