域上保持m×n秩1矩阵的函数

Functions preserving m×n rand-1 matrices of order n over fields

设F是任意的域,m,n是整数,m,n≥2.对于一个函数f:F→F和F上的一个矩阵A=[aij],用符号Af定义矩阵[f(aij)].如果秩Af=1对F上所有的m×n秩1矩阵A成立,则称f保持m×n秩1矩阵.刻画了F上所有保持m×n秩1矩阵的函数的一般形式.这推广了最近的文献Kalinowski[1,2]中的结论.

Let F be an arbitrary field and m, n be integers with m, n≥2. For a function f： F→F and a matrix A = [aij] over F,let A^f be the matrix [f（aij）]. If rank A^f = 1 for all m ×n rank-1 matrices,then f is called a function preserving m × n rank-1 matrices. The general form of all functions preserving m × n rank- 1 matrices over F is characterized. This generalizes the recent results of Kalinowski[ 1,2].