$$$$ \begin{bmatrix} \ 0 & 1 & 2 & 1 \\ \ \ 1 & 2 & 3 & 2 \\ \ \ 2 & 2 & 4 & 3 \\ \ \ 1 & 2 & 3 & 2 \\ \end{bmatrix} $$$$

By 2D DFT,

$F(u,v) = \frac{1}{N} \sum_{x=0}^{N-1} \sum_{y=0}^{N-1} f(x,y) W_N^{xu} W_N^{yv}$

In matrix form, $T=AFA^T$

$T = \frac{1}{\sqrt{4}} \begin{bmatrix} \ 1 & 1 & 1 & 1 \\ \ 1 & -j & -1 & j \\ \ 1 & -1 & 1 & -1 \\ \ 1 & j & -1 & -j \\ \end{bmatrix} \begin{bmatrix} \ 0 & 1 & 2 & 1 \\ \ 1 & 2 & 3 & 2 \\ \ 2 & 3 & 4 & 3 \\ \ 1 & 2 & 3 & 2 \\ \end{bmatrix} A^T$

$T = \frac{1}{\sqrt{4}} \begin{bmatrix} \ 4 & 8 & 12 & 8 \\ \ -2 & -2 & -2 & -2 \\ \ 0 & 0 & 0 & 0 \\ \ -2 & -2 & -2 & -2 \\ \end{bmatrix} \frac{1}{\sqrt{4}} \begin{bmatrix} \ 1 & 1 & 1 & 1 \\ \ 1 & -j & -1 & j \\ \ 1 & -1 & 1 & -1 \\ \ 1 & j & -1 & -j \\ \end{bmatrix}$

$T = \frac{1}{4} \begin{bmatrix} \ 32 & -8 & 0 & -8 \\ \ -8 & 0 & 0 & 0 \\ \ 0 & 0 & 0 & 0 \\ \ -8 & 0 & 0 & 0 \\ \end{bmatrix}$

$T = \begin{bmatrix} \ 8 & -2 & 0 & -2 \\ \ -2 & 0 & 0 & 0 \\ \ 0 & 0 & 0 & 0 \\ \ -2 & 0 & 0 & 0 \\ \end{bmatrix}$