Different Dimension Of Space Linear Preserver Problems

Let F be a field of characteristic not 2 or 3 , C be a complex field. Let Mn(F) (or Mn(C)) be the vector space of all n × n matrices over F(or C), Sn (F) (or Sn(C)) be the vector space of n×n symmetric matrices over F(or C).We characterize the linear maps preserving inverse from Sn(F) to Mm(F) and from Sn(F) to Sm(F). We also characterize the linear maps preserving k-power from Mn(C) to Mm(C) and from Sn(C) to Mm(C) .