Discrete Fourier transform with matrix

Question

Kindly help or point me in the right direction. All research and videos I have seen all use linear values.

Discrete Fourier transform question

Answer 1

Let me say something about discrete Fourier transforms. Let $N\ge 1$ and define $\omega = e^{-\frac{2\pi i}{N}}$. If you already took a course on complex analysis, you recognize $\omega^{k} = e^{-\frac{2\pi i k}{N}}$, with $k=0,1,...,N-1$ as the complex roots of $1$. Now, the Fourier transform of order $N$ is defined by means of the following matrix: $$ F_{N} := \frac{1}{\sqrt{N}} \begin{pmatrix} 1 & 1 & \cdots & 1 \\ 1 & \omega & \cdots & \omega^{N-1} \\ 1 & \omega^{2} & \cdots & \omega^{2(N-1)} \\ \vdots & \vdots & \ddots &\vdots \\ 1 & \omega^{N-1} & \cdots & \omega^{(N-1)(N-1)} \end{pmatrix} \hspace{1cm} (2) $$ Here, $F_{N}$ is the so-called Fourier matrix of $N$-th order. Finally, if you have a complex vector $z = (z_{1},...,z_{N})$, its Fourier transform is defined to be: $$\hat{z} := F_{N}z \hspace{1cm} (2)$$ where I'm assuming matrix multiplication, obviously.

With all these being said, you have to compute the Fourier transform of a vector $z= (2, -2-2i, -2i, 4+4i)$. The discussion above indicates that you should use the $4$-th order Fourier matrix. So, your task is simply to construct $F_{4}$ following the prescription given by (1) and then apply (2) to your vector $z$. Can you follow from here?

Note: For further discussions about this subject, you can check out Davis' book or chapter 7 of Stein's book.