  Let $X_1,X_2, \ldots $ be a sequence of $i.i.d$ real (complex) $d \times d $
invertible random matrices with common distribution $\mu$ and $\sigma_1(n),
\sigma_2(n), \ldots , \sigma_d(n)$ be the singular values, $\lambda_1(n),
\lambda_2(n), \ldots , \lambda_d(n)$ be the eigenvalues of $X_nX_{n-1}\cdots
X_1$ in the decreasing order of their absolute values for every $n$. It is
known that if $\mathbb{E}(\log^{+}\|X_1\|)< \infty$, then with probability one
for all $1 \leq p \leq d$, $$
  \lim_{n \to \infty} \frac{1}{n}\log \sigma_p(n)=\gamma_p, $$ where
${\gamma_1,\gamma_2 \ldots \gamma_d}$ are the Lyapunov exponents associated
with $\mu$. In this paper we show that under certain support and moment
conditions on $\mu$, the absolute values of eigenvalues also exhibit the same
asymptotic behaviour. In fact, a stronger asymptotic relation holds between the
singular values and the eigenvalues $i.e.$ for any $r>0$ with probability one
for all $1 \leq p \leq d$, $$ \lim_{n \to \infty} \frac{1}{n^r}\log
\left(\frac{|\lambda_p(n)|}{\sigma_p(n)}\right)= 0, $$ which implies that the
fluctuations of the eigenvalues have the same asymptotic distribution as that
of the corresponding singular values. Isotropic random matrices and also random
matrices with $i.i.d$ real elements, which have some finite moment and bounded
density whose support contains an open set, are shown to satisfy the moment and
support conditions under which the above relations hold.
