in 10.6.2 it is said that it can be easily shown that (A^T)A = I, but since A^T is exactly (V^T)(X^T)

(since a_i = (V^T)x_i) then (A^T)A = (V^T)(X^T)XV. we know that the solution V is a matrix where the i'th column is the eigenvector u_i with eigenvalue lambda_i (of (X^T)X) and therefore (V^T)(X^T)XV = (V^T)Z where Z is a matrix with lambda_i*u_i as the i'th column and finally (V^T)Z = Diag(lambda_1, … , lambda_r) since V is orthonormal, and not I_r as pointed in the scribe.

Am I missing anything?

Thanks!