The Inverse DFT

The key to seeing that the inverse of the DFT is itself a DFT is to see that although it is not computed as a matrix product, the result of the DFT is a matrix product.   

As we have seen,   If we look at any specific component we see

        

So the complete output is


        

where is the Vandermonde matrix

which has in its row, column

Clearly, if we know an output , we can recover the value of a from the equation
            

Theorem.  If ω is a primitive   root of unity and   is the corresponding Vandermonde matrix, then

            

Proof:  We simply compute the product and verify that it is equal to    This implies that   =N V[ω] .   The factor V[ω] can be cancelled and then the conclusion of the theorem follows.  As for the product:



and if

Examples.
We saw above that the complex number is a primitive root of unity and

In the field of integers modulo 17, 9 is a primitive 8th root of unity and 2 is its inverse.

Back to inverting the DFT.   Since inverting the DFT is the same as multiplying by the a Vandermonde matrix, we can conclude that

        If ,  then