  Consider a branching random walk $(G_u)_{u\in \mathbb T}$ on the general
linear group $\textrm{GL}(V)$ of a finite dimensional space $V$, where $\mathbb
T$ is the associated genealogical tree with nodes $u$. For any starting point
$v \in V \setminus\{0\}$, let $M^v_n=\max_{|u| = n} \log \| G_u v \|$ denote
the maximal position of the walk $\log \| G_u v \|$ in the generation $n$. We
first show that under suitable conditions, $\lim_{n \to \infty} \frac{M_n^v
}{n} = \gamma_{+}$ almost surely, where $\gamma_{+}\in \mathbb R$ is a
constant. Then, in the case when $\gamma_+=0$, under appropriate boundary
conditions, we refine the last statement by determining the rate of convergence
at which $M_n^v$ converges to $-\infty$. We prove in particular that $\lim_{n
\to \infty} \frac{M_n^v}{\log n} = -\frac{3}{2\alpha}$ in probability, where
$\alpha >0$ is a constant determined by the boundary conditions. Similar
properties are established for the minimal position. As a consequence we derive
the asymptotic speed of the maximal and minimal positions for the coefficients,
the operator norm and the spectral radius of $G_u$.
