  In the present paper, we treat random matrix products on the general linear
group $\textrm{GL}(V)$, where $V$ is a vector space defined on any local field,
when the top Lyapunov exponent is simple, without irreducibility assumption. In
particular, we show the existence and uniqueness of the stationary measure
$\nu$ on $\textrm{P}(V)$ that is relative to the top Lyapunov exponent and we
describe the projective subspace generated by its support. We observe that the
dynamics takes place in a open set of $\textrm{P}(V)$ which has the structure
of a skew product space. Then, we relate this support to the limit set of the
semi-group $T_{\mu}$ of $\textrm{GL}(V)$ generated by the random walk.
Moreover, we show that $\nu$ has H\"older regularity and give some limit
theorems concerning the behavior of the random walk and the probability of
hitting a hyperplane. These results generalize known ones when $T_{\mu}$ acts
strongly irreducibly and proximally (i-p to abbreviate) on $V$. In particular,
when applied to the affine group in the so-called contracting case or more
generally when the Zariski closure of $T_{\mu}$ is not necessarily reductive,
the H\"older regularity of the stationary measure together with the description
of the limit set are new. We mention that we don't use results from the i-p
setting; rather we see it as a particular case.
