  There is a natural epimorphism from the symmetric algebra to the Rees algebra
of an ideal. When this epimorphism is an isomorphism, we say that the ideal is
of linear type. Given two determinantal rings over a field, we consider the
diagonal ideal, the kernel of the multiplication map. We prove that the
diagonal ideal is of linear type and recover the defining ideal of the Rees
algebra in some special cases. The special fiber ring of the diagonal ideal is
the homogeneous coordinate ring of the join variety.
