Are the real parts of the vectors constituting the discrete Fourier transform matrix linearly independent?

Question

Let W denote the n- dimensional symmetric discrete Fourier transform matrix and $W_{i}$ denote its column vectors. Then, is the set { Re($W_{i}$) | i= 1... n } linearly independent? Or similarly, find det( Re ( W ) ).

Answer 1

The short answer is: No.

For example, consider $n=3$. You have $\omega=e^{-2\pi i/ 3} = -\frac{1}{2} - i \frac{\sqrt{3} }{2}$.

Then; the corresponging DTF matrix, $W$, becomes $$ W= \begin{pmatrix} 1 & 1 & 1 \\ 1 & \omega &\omega^2 \\ 1 & \omega^2 & \omega^4\\ \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & e^{-2\pi i/ 3} &e^{-4\pi i/ 3} \\ 1 & e^{-4\pi i/ 3} & e^{-8\pi i/ 3}\\ \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & -\frac{1}{2} - i \frac{\sqrt{3}}{2} &-\frac{1}{2} + i \frac{\sqrt{3}}{2} \\ 1 & -\frac{1}{2} + i \frac{\sqrt{3}}{2} & -\frac{1}{2} - i \frac{\sqrt{3}}{2}\\ \end{pmatrix}. $$ Giving: $$\Re(W)= \begin{pmatrix} 1 & 1 & 1 \\ 1 & -1/2 &-1/2 \\ 1 & -1/2 & -1/2\\ \end{pmatrix}. $$

And the set $ \{ \, \Re(W_i) \mid i= 1,2,3 \, \} = \Bigg\{ \begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix}, \begin{pmatrix} 1 \\ -1/2 \\ -1/2 \\ \end{pmatrix}, \begin{pmatrix} 1 \\ -1/2 \\ -1/2 \\ \end{pmatrix} \Bigg\} $ is not linearly independent since the last two vectors are equal.

Moreover; we can prove that for each $n\geq3$, the column vectors of the real part of the corresponding DFT matrix $W$ are in fact linearly dependent! A simple proof is as follows:

$$ \Re(W)_2 =\Re(W_2) = \Re \Big(\begin{pmatrix} 1 \\ \omega \\ \omega^2 \\ \vdots\\ \omega^{n-1} \end{pmatrix}\Big) = \Re \Big(\begin{pmatrix} 1 \\ \omega^{-1} \\ \omega^{-2} \\ \vdots\\ \omega^{-(n-1)} \end{pmatrix}\Big) = \Re \Big(\begin{pmatrix} 1 \\ \omega^{-1} \times \omega^n\\ \omega^{-2} \times \omega^{2n} \\ \vdots\\ \omega^{-(n-1)} \times \omega^{(n-1)n} \end{pmatrix} \Big) = \Re \Big(\begin{pmatrix} 1 \\ \omega^{n-1} \\ \omega^{2(n-1)} \\ \vdots\\ \omega^{(n-1)(n-1)} \end{pmatrix} \Big) =\Re (W_n) =\Re (W)_n $$ since $\omega^{nk}=1$ for any integer $k$.

Thus, the second and n-th columns of $\Re(W)$ are equal if $n\geq3$; which, in the end shows that the column vectors are not linearly independent.