Let \(M_n(\mathbb {F})\) be the algebra of \(n \times n\) matrices over a field \(\mathbb {F}\) of characteristic not equal to 2. If \(n\ge 2\) , we show that an arbitrary map \(\phi : M_n(\mathbb {F}) \rightarrow M_n(\mathbb {F})\) is Jordan multiplicative, i.e. it satisfies the functional equation \(\begin{aligned} \phi (XY+YX)=\phi (X)\phi (Y)+\phi (Y)\phi (X), \quad \text {for all } X,Y \in M_n(\mathbb {F}) \end{aligned}\) if and only if one of the following holds: either \(\phi \) is constant and equal to a fixed idempotent, or there exists an invertible matrix \(T \in M_n(\mathbb {F})\) and a ring monomorphism \(\omega : \mathbb {F} \rightarrow \mathbb {F}\) such that \(\begin{aligned} \phi (X)=T\omega (X)T^{-1} \quad \text { or } \quad \phi (X)=T\omega (X)^tT^{-1}, \quad \text {for all } X \in M_n(\mathbb {F}), \end{aligned}\) where \(\omega (X)\) denotes the matrix obtained by applying \(\omega \) entrywise to X. In particular, any Jordan multiplicative map \(\phi : M_n(\mathbb {F}) \rightarrow M_n(\mathbb {F})\) with \(\phi (0)=0\) is automatically additive. The analogous characterization fails when \({\mathbb {F}}\) has characteristic 2.
