### Hermitian nonnegative-definite and positive-definite solutions of the matrix equation $AXB=C$.

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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...

We extend three inequalities involving the Hadamard product in three ways. First, the results are extended to any partitioned blocks Hermitian matrices. Second, the Hadamard product is replaced by the Khatri-Rao product. Third, the necessary and sufficient conditions under which equalities occur are presented. Thereby, we generalize two inequalities involving the Khatri–Rao product.

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