  In this paper, we give a complete self-contained proof that the rigidity
matrix of a symmetric bar and joint framework (as well as its transpose) can be
transformed into a block-diagonalized form using techniques from group
representation theory. This theorem is basic to a number of useful and
interesting results concerning the rigidity and flexibility of symmetric
frameworks. As an example, we use this theorem to prove a generalization of the
Fowler-Guest symmetry extension of Maxwell's rule which can be applied to both
injective and non-injective realizations in all dimensions.
