域上矩阵保乘积的诱导映射

Induced Maps Preserving Multiplicative Matrices over Fields

摘要:

令F是一个域，S_n(F)是F上所有n*n对称矩阵的集合。如果一个映射f:S_n(F)→S_n(F)被定义如下，∫:B=(b_ij)|→(f_ij(b_ij)), ∀B∈S_n(F)。

其中，{f_ij|i≤j∈{1,2,...,n}}是关于F的函数集，则称f是S_n(F)的由{f_ij}诱导的映射。如果对于A,B∈S_n(F)有f(AB)=f(A)f(B)，则f被称为保矩阵乘积。本文我们刻画域上矩阵保乘积的诱导映射。

Abstract: Let F be a field, S_n(F) be the set of all n*n matrices over F. If a map f:S_n(F)→S_n(F) is defined by ∫:B=(b_ij)|→(f_ij(b_ij)) where {f_ij|i≤j∈{1,2,...,n}} is the set of functions on F, then f is called a map induced by {f_ij} on S_n(F). If A,B∈S_n(F) implies f(AB)=f(A)f(B), then f is called preserving multiplicative matrices. In this paper, we characterize induced maps preserving multiplicative matrices over fields.

文章引用：张隽, 曹重光. 域上矩阵保乘积的诱导映射[J]. 理论数学, 2016, 6(3): 166-171. http://dx.doi.org/10.12677/PM.2016.63025

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