上三角矩阵的线性交换映射的表示

设Trn(R)表示定义在实数域R上的n×n阶上三角矩阵的集合,φ是定义Trn(R)上线性映射.如果对任意X∈Trn(R)有Xφ(X)=φ(X)X成立,称φ是线性交换映射.本文利用初等的矩阵计算方法描述了当φ(I)=I时,线性交换映射φ的表示形式,而且给出了φ的Frobenius范数‖φ(X)‖F的估计.

The Representation of Linear Commuting Map on Upper Triangular Matrix

Let Trn(R) denote the set of n×n upper triangular matrices over real field R,φ be a linear map from Trn(R) to Trn(R) if Xφ(X)=φ(X)X for arbitrary X∈Trn(R),φ is said to be commuting.In this paper,we describe the representation of φ satisfying φ(I)=I by elementary method of matrix computation,and give the estimate of Frobenius norm of φ(X).