I'm trying to make a gradient flow for an image. For a test I made a small image 3x3 pixels with a black pixel in the middle.

I found how to compute the direction of the gradient for one point given by coordinates (x,y)

$$\varphi (x,y)=arctan\frac{G_{y}}{G_{x}}$$

Using Sobel operator I got these gradients: $$G_x = \begin{bmatrix} 255 & 0 & -255 \\ 510 & 0& -510\\ 255 & 0 & -255 \end{bmatrix}, G_y = \begin{bmatrix} 255 & 510 & 255 \\ 0 & 0& 0\\ -255 & -510 & -255 \end{bmatrix}$$

I also computed the gradient magnitude:

$$\left \| G\right \|^{2} = \begin{bmatrix} 130050.00 & 260100.00 & 130050.00\\ 260100.00& 0 & 260100.00\\ 130050.00& 260100.00 &130050.00 \end{bmatrix}$$

but I have no idea how to make an image with the arrows representing the direction of the gradient. Could anyone please explain me how to do it? Thanks.

No need for the $\arctan$ or $|| \cdot ||^2$. The coordinates of your arrow are directly given by $G_x$ and $G_y$. Also the expression for the square nom is a bit suspicious since it is not simply a convolution.