ELA, Volume 11, pp. 205-211, September 2004, abstract.

Determinant Preserving Transformations on Symmetric 
Matrix Spaces

Chongguang Cao and Xiaomin Tang

Let S_n(F) be the vector space of n-by-n symmetric
matrices over a field F (with certain restrictions on
cardinality and characteristic). The transformations
phi on the space which satisfy one of the following 
conditions:

1. det(A+ lambda B)= det(phi(A)+lambda phi(B)) 
   for all A, B in S_n(F) and lambda in F;

2. phi is surjective and 
   det(A+ lambda B)=det(phi(A)+ lambda phi(B)) 
   for all A, B and two specific lambda;

3. phi is additive and preserves determinant

are characterized.