On linear transformations preserving at least one eigenvalue

Let $F$ be an algebraically closed field and $T: M_n(F) \longrightarrow M_n(F)$ be a linear transformation. In this paper we show that if $T$ preserves at least one eigenvalue of each matrix, then $T$ preserves all eigenvalues of each matrix. Moreover, for any infinite field $F$ (not necessarily algebraically closed) we prove that if $T: M_n(F) \longrightarrow M_n(F)$ is a linear transformation and for any $A\in M_n(F)$ with at least an eigenvalue in $F$, $A$ and $T(A)$ have at least one common eigenvalue in $F$, then $T$ preserves the characteristic polynomial.