Orthogonality and Inverse of DFT Matrix

Question

Since there are missing information about my question, I re-open the topic. I am trying to proof that the DFT matrix is orthogonal but i stucked. I just know to proof the orthogonality of two vectors, not a single matrix. How can we proof that the DFT matrix is orthogonal? Thanks.

Answer 1

Notation I'm using is the one used here.

I'm not assuming that the entries of the matrix are properly normalized :

Let's called the matrix $\Omega_n$. The matrix is symmetric since $\omega_n^{ij} = \omega_n^{ji}$. To prove the fact about orthogonality on the columns of the matrix we have to show that the product between the $i$-th row of $\Omega_n^H$ and the $j$-th column of $\Omega_n$ is $0$ if $i \ne j$ and $n$ if $i=j$, i.e $$\sum\limits_{k=0}^{n-1} \bar{\omega_n}^{ik}\omega_n^{jk} = \begin{cases}n & i = j \\ 0 & i \ne j \end{cases}$$

The sum reduces to $\sum\limits_{k=0}^{n-1}\omega_n^{rk}$ where $r = j-i$. So the thesis follows from :

Lemma : $\sum\limits_{i=0}^{n-1} \omega_n^{ki} = \begin{cases} n & k \equiv 0 \hspace{0.1cm} \text{mod} \hspace{0.1cm} n \\ 0 & k \not\equiv 0 \hspace{0.1cm} \text{mod} \hspace{0.1cm} n \end{cases}$