Determinant preserving transformations on symmetric matrix spaces

Abstract

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

1. det(A + λB) = d et(φ(A) + λφ(B)) for all A,B ∈ Sn(F) and λ ∈ F;

2. φ is surjective and det(A + λB) = det(φ(A) + λφ(B)) for all A, B and two specific λ;

3. φ is additive and preserves determinant

are characterized.