# Rigid Deformation

I'm trying to parse through this paper on using the method of moving least squares for rigid transformations - http://www.cs.rice.edu/~jwarren/research/mls.pdf

Under section 2.3, the author mentions that the optimal rigid transformation can be found in terms of eigenvalues and eigenvectors of a covariance matrix involving the points $p_i$ and $q_i$, which represent the original and deformed points respectively.

One thing I don't understand is how the covariance matrix is constructed and how one would go about getting the optimal rigid transformation from it. Could anyone shed some light on what this means?