Determinants Preserving Maps on the Spaces of Symmetric Matrices and Skew-Symmetric Matrices

Abstract

Denote by Σ_n and Q_n the set of all n × n symmetric and skew-symmetric matrices over a field $\mathbb {F}$, respectively, where $\text {char}(\mathbb {F})\neq 2$ and $|\mathbb {F}| \geq n^{2}+1$. A characterization of $\phi ,\psi :{\varSigma }_{n} \rightarrow {\varSigma }_{n}$, for which at least one of them is surjective, satisfying

$ \det (\phi (x)+\psi (y))=\det (x+y)\qquad (x,y\in {\varSigma }_{n}) $

is given. Furthermore, if n is even and $\phi ,\psi :Q_{n} \rightarrow Q_{n}$, for which ψ is surjective and ψ(0) = 0, satisfy

$\det (\phi (x)+\psi (y))=\det (x+y)\qquad (x,y\in Q_{n}), $

then ϕ = ψ and ψ must be a bijective linear map preserving the determinant.