  Let $(g_{n})_{n\geq 1}$ be a sequence of independent identically distributed
$d\times d$ real random matrices with Lyapunov exponent $\gamma$. For any
starting point $x$ on the unit sphere in $\mathbb R^d$, we deal with the norm $
| G_n x | $, where $G_{n}:=g_{n} \ldots g_{1}$. The goal of this paper is to
establish precise asymptotics for large deviation probabilities $\mathbb P(\log
| G_n x | \geq n(q+l))$, where $q>\gamma $ is fixed and $l$ is vanishing as
$n\to \infty$. We study both invertible matrices and positive matrices and give
analogous results for the couple $(X_n^x,\log | G_n x |)$ with target
functions, where $X_n^x= G_n x /| G_n x |$. As applications we improve previous
results on the large deviation principle for the matrix norm $\|G_n\|$ and
obtain a precise local limit theorem with large deviations.
