  Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian
matrices, symmetric matrices, positive definite (resp. semi-definite) matrices,
diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus
\{0, 1\}$. We characterize linear maps $\psi:V\to V$ that satisfy
$\psi(A^k)=\psi(A)^k$ on an open neighborhood $S$ of $I_n$ in $V$. The
$k$-power preservers are necessarily $k$-potent preservers, and the case $k=2$
corresponds to Jordan homomorphisms. Applying the results, we characterize maps
$\phi,\psi:V\to V$ that satisfy "$
\operatorname{tr}(\phi(A)\psi(B)^k)=\operatorname{tr}(AB^k)$ for all $A\in V$,
$B\in S$, and $\psi$ is linear" or "$
\operatorname{tr}(\phi(A)\psi(B)^k)=\operatorname{tr}(AB^k)$ for all $A, B\in
S$ and both $\phi$ and $\psi$ are linear." The characterizations systematically
extend existing results in literature, and they have many applications in areas
like quantum information theory. Some structural theorems and power series over
matrices are widely used in our characterizations.
