对称矩阵空间上保逆线性映射

Linear maps preserving inverses of matrices on symmetric matrix space

设F是特征不为2且元素个数大于3的域,n和m是正整数,令Sn(F)和Mn(F)分别是F上n×n对称矩阵空间和全矩阵空间,GLm(F)为F上m阶一般线性群,设f是从Sn(F)到Mm(F)上的线性映射,若f满足f(X)-1=f(X-1),X∈Sn(F)∩GLn(C),称f为保逆线性映射.刻画了从Sn(F)到Mm(F)以及从Sn(F)到Sm(F)上保逆线性映射.

Suppose F is a field of characteristic not 2 and ｜ F｜ 〉 3, m and n are positive integers, Let Sn （F） and Mm, （F） be the vector spaces of all n×n symmetric matrices and all m×m full matrices over F, respectively. Let GLn （F） be the set of all n x n nonsingular matrices. A linear map f from Sn （F） to Mm （F） is said to inverse - preserving if f （X）^-1 =f（X^-1） for every X∈Sn （F） ∩GLn （F）. Linear maps preserving inverses of matrices from Sn（F） to Mm（F） （ respectively Sm（F） ） are characterized.