  Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically
distributed elements of the general linear group $GL(d, \mathbb R)$. Consider
the random walk $G_n: = g_n \ldots g_1$. Under suitable conditions, we
establish Bahadur-Rao-Petrov type large deviation expansion for the
coefficients $\langle f, G_n v \rangle$, where $f \in (\mathbb R^d)^*$ and $v
\in \mathbb R^d$. In particular, our result implies the large deviation
principle with an explicit rate function, thus improving significantly the
large deviation bounds established earlier. Moreover, we establish
Bahadur-Rao-Petrov type large deviation expansion for the coefficients $\langle
f, G_n v \rangle$ under the changed measure. Toward this end we prove the
H\"{o}lder regularity of the stationary measure corresponding to the Markov
chain $G_n v /|G_n v|$ under the changed measure, which is of independent
interest. In addition, we also prove local limit theorems with large deviations
for the coefficients of $G_n$.
