矩阵空间上秩加性的加法保持

设IF是域,V是或者域IF上所有m×n矩阵的空间或者是特征不为2及3的域IF上所有n×n对称矩阵的空间.对于每个被固定的正整数s≥2,Qs定义V×V中满足rank(A+B)=rank(A)+rank(B)≤s的所有矩阵对(A,B)的集合.刻划了V上满足ψ(Qs)(∈)Qs的加法映射ψ.当charIF≠2时,也描述了IF上从n×n矩阵空间到p×q矩阵空间保秩加性的线性算子的结构.

Additive preservers of rank-additivity on matrix spaces

Let IF be a field. Let V denote the vector space of all m × n matrices over F or the vector space of all n × n symmetric matrices over IF of characteristic not 2 or 3. For each fixed positive s ≥ 2, let Qs denote the set of all matrix pairs (A, B) in V such that rank(A + B) = rank(A) + rank(B) ≤ s. The authors characterize additive mappings ψ on V × V such that ψ(Qs)(∈)Qs for a fi×ed s. They also describe the structure of linear mappings from the space of n × n matrices over IF to space of p × q matrices over IF that preserve rank-additivity, where char IF ≠ 2.