  Ensembles of isotropic random matrices are defined by the invariance of the
probability measure under the left (and right) multiplication by an arbitrary
unitary matrix. We show that the multiplication of large isotropic random
matrices is spectrally commutative and self-averaging in the limit of infinite
matrix size $N \rightarrow \infty$. The notion of spectral commutativity means
that the eigenvalue density of a product ABC... of such matrices is independent
of the order of matrix multiplication, for example the matrix ABCD has the same
eigenvalue density as ADCB. In turn, the notion of self-averaging means that
the product of n independent but identically distributed random matrices, which
we symbolically denote by AAA..., has the same eigenvalue density as the
corresponding power A^n of a single matrix drawn from the underlying matrix
ensemble. For example, the eigenvalue density of ABCCABC is the same as of
A^2B^2C^3. We also discuss the singular behavior of the eigenvalue and singular
value densities of isotropic matrices and their products for small eigenvalues
$\lambda \rightarrow 0$. We show that the singularities at the origin of the
eigenvalue density and of the singular value density are in one-to-one
correspondence in the limit $N \rightarrow \infty$: the eigenvalue density of
an isotropic random matrix has a power law singularity at the origin $\sim
|\lambda|^{-s}$ with a power $s \in (0,2)$ when and only when the density of
its singular values has a power law singularity $\sim \lambda^{-\sigma}$ with a
power $\sigma = s/(4-s)$. These results are obtained analytically in the limit
$N \rightarrow \infty$. We supplement these results with numerical simulations
for large but finite N and discuss finite size effects for the most common
ensembles of isotropic random matrices.
