  A random matrix with rows distributed as a function of their length is said
to be isotropic. When these distributions are Gaussian, beta type I, or beta
type II, previous work has, from the viewpoint of integral geometry, obtained
the explicit form of the distribution of the determinant. We use these result
to evaluate the sum of the Lyapunov spectrum of the corresponding random matrix
product, and we further give explicit expressions for the largest Lyapunov
exponent. Generalisations to the case of complex or quaternion entries are also
given. For standard Gaussian matrices $X$, the full Lyapunov spectrum for
products of random matrices $I_N + {1 \over c} X$ is computed in terms of a
generalised hypergeometric function in general, and in terms of a single single
integral involving a modified Bessel function for the largest Lyapunov
exponent.

Probability