Let F be an algebraically closed field and T : Mn(F) −→ Mn(F)be a linear transformation. In this paper we show that if T preserves atleast one eigenvalue of each matrix, then T preserves all eigenvalues of eachmatrix. Moreover, for any infinite field F (not necessarily algebraically closed)we prove that if T : Mn(F) −→ Mn(F) is a linear transformation and for anyA ∈ Mn(F) with at least an eigenvalue in F, A and T(A) have at least onecommon eigenvalue in F, then T preserves the characteristic polynomial.