I don't know how you got that ||x|| = 1 is a critical point. I got that either lambda or x for each i must be 0 (one or the other or both must be zero), so then Dx = 0 because Dx = the sum of lambda times x for each i. Since one of those two values is zero for all i, then that will always be zero.
I justified changing back because changing Dx back to the original basis involves matrix multiplication only, which 0 times and matrix is still 0. Thus Ax = 0 as well.