Let $M_n$ be the multiplicative semigroup of all $n\times n$ complex matrices, and let $U_n$ and $G{L}_n$ be the $n$–degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from $U_n$ to $G{L}_m$ when $n>m\ge 1$ or $n=m\ge 3$, and thereby determine multiplicative homomorphisms from $U_n$ to $M_m$ when $n>m\ge 1$ or $n=m\ge 3$. This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if $f:U_n\to M_n$ is a spectrum–preserving multiplicative homomorphism, then there exists a matrix $R$ in $G{L}_n$ such that $f\left(A\right)=RAR$ for...