  Using the theory of equitable decompositions it is possible to decompose a
matrix $M$ appropriately associated with a given graph. The result is a
collection of smaller matrices whose collective eigenvalues are the same as the
eigenvalues of the original matrix $M$. This is done by decomposing the matrix
over a graph symmetry. Previously it was shown that a matrix can be equitably
decomposed over any uniform, basic, or separable automorphism. Here we extend
this theory to show that it is possible to equitably decompose a matrix over
any automorphism of a graph, without restriction. Moreover, we give a
step-by-step procedure which can be used to generate such a decomposition. We
also 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
$M$.
