  We prove a central limit theorem for the components of the eigenvectors
corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix
of a finite dimensional random dot product graph. As a corollary, we show that
for stochastic blockmodel graphs, the rows of the spectral embedding of the
normalized Laplacian converge to multivariate normals and furthermore the mean
and the covariance matrix of each row are functions of the associated vertex's
block membership. Together with prior results for the eigenvectors of the
adjacency matrix, we then compare, via the Chernoff information between
multivariate normal distributions, how the choice of embedding method impacts
subsequent inference. We demonstrate that neither embedding method dominates
with respect to the inference task of recovering the latent block assignments.
