可换对称矩阵上的非线性k次幂等保持映射(英文)

Non-linear k-potence preserving maps on commuting symmetric matrices

线性保持问题主要研究矩阵空间上保持某种函子、子集合或者某种关系式等不变的算子.研究了复数域上对称矩阵空间的非线性保持问题,运用矩阵计算技巧和数学归纳法,证明了可换对称矩阵组A=(A1,A2,…,Ad)上保持k次幂等的非线性映射是一个k次单位根与一个依赖于A的内自同构的乘积.这一结论是一些已知结果的重要补充.

Linear preserver problems mainly investigate the operators on matrix spaces that leave certain functions,subsets,relations,etc.,invariant.Nonlinear preserver problem on symmetric matrices over complex field is investigated in the present paper.By using matrix computational techniques and induction,it is shown that the nonlinear maps on commuting symmetric matrices A=（A1,A2,…,Ad）that preserve k-potent is a product of a k-th root of unity and an inner automorphism depending on A.The conclusion is an important supplement of some known results.