  If $A$ and $B$ are nonnegative matrices, a sharp upper bound on the spectral
radius $\rho(A\circ B)$ for the Hadamard product of two nonnegative matrices is
given, and the minimum eigenvalue $\tau(A\star B)$ of the Fan product of two
$M$-matrices $A$ and $B$ is discussed. In addition, we also give a sharp lower
bound on $\tau(A\circ B^{-1})$ for the Hadamard product of $A$ and $B^{-1}$.
Several examples, illustrating that the given bound is stronger than the
existing bounds, are also given.
