Quantum fourier transformation Unitary proof.

Question

I've found a bunch of these proofs online but I am having trouble understanding how the norm of the column/row is 1.

Answer 1

If $i=j$ you get $$FT_N \cdot FT_N = \frac1N \sum_{k=0}^{N-1} \omega^{ik - ik} = \frac1N \sum_{k=0}^{N-1} 1 = 1$$ Note that $\sum_{k\in\mathbb Z_N}$ is the same as $\sum_{k=0}^{N-1}$. This is due to the definition of $\omega$ (actually dependent on $N$): $$\omega = e^{\frac{2\pi i}N}$$ So that $\omega^k = e^{\frac{2\pi ik}N}$ and $\bar\omega = \omega^{-1}$ so $\omega^i \bar\omega^j = \omega^{i-j}$