  We extend the theory of equitable decompositions, in which, if a graph has a
particular type of symmetry, i.e. a uniform or basic automorphism $\phi$, it is
possible to use $\phi$ to decompose a matrix $M$ appropriately associated with
the graph. The result is a number of strictly smaller matrices whose collective
eigenvalues are the same as the eigenvalues of the original matrix $M$. We show
here that a large class of automorphisms, which we refer to as
\emph{separable}, can be realized as a sequence of basic automorphisms,
allowing us to equitably decompose $M$ over any such automorphism. We also show
that not only can a matrix $M$ be decomposed but that the eigenvectors of $M$
can also be equitably decomposed. Additionally, we prove under mild conditions
that if a matrix $M$ is equitably decomposed the resulting divisor matrix,
which is the divisor matrix of the associated equitable partition, will have
the same spectral radius as the original matrix $M$. Last, we describe how an
equitable decomposition effects the Gershgorin region $\Gamma(M)$ of a matrix
$M$, which can be used to localize the eigenvalues of $M$. We show that the
Gershgorin region of an equitable decomposition of $M$ is contained in the
Gershgorin region $\Gamma(M)$ of the original matrix. We demonstrate on a
real-world network that by a sequence of equitable decompositions it is
possible to significantly reduce the size of a matrix' Gershgorin region.
