  In this paper, we study an analytic curve $\varphi: I=[a,b]\rightarrow
\mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices,
and show that if $\varphi$ satisfies certain geometric condition, then for
almost every point on the curve, the Diophantine approximation given by
Dirichlet's Theorem can not be improved. To do this, we embed the curve into
some homogeneous space $G/\Gamma$, and prove that under the action of some
expanding diagonal subgroup $A= \{a(t): t \in \mathbb{R}\}$, the translates of
the curve tend to be equidistributed in $G/\Gamma$, as $t \rightarrow +\infty$.
