  We consider orthogonally invariant probability measures on
$\mathrm{GL}_n(\mathbb{R})$ and compare the mean of the logs of the moduli of
eigenvalues of the matrices to the Lyapunov exponents of random matrix products
independently drawn with respect to the measure. We give a lower bound for the
former in terms of the latter. The results are motivated by
Dedieu-Shub\cite{DS}. A novel feature of our treatment is the use of the theory
of spherical polynomials in the proof of our main result.
