Compute the two-dimensional DFT

Compute the two-dimensional DFT [4x4] for the following 4x4 image $$\begin{matrix} 0.5 & 0.5 & 0.5 & 0.5 \\ 0.5 & 0.5 & 0.5 & 0.5\\ 0.5 & 0.5 & 0.5 & 0.5\\ 0.5 & 0.5 & 0.5 & 0.5 \end{matrix}$$

I know that DFT is separable by dimensions – one can calculate 4 vertical transforms first, then 4 horizontal ones

For each row we get [2 0 0 0] in the first and third row and zero elsewhere.

For each column we get [1 0 1 0] in the first and third column and zero elsewhere.

How from this two one-dimensional dfts obtain two-dimensional one?

$$H_1 = \begin{matrix} 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ \end{matrix}$$
and the vertical 1D-DFT of the columns of $$H_1$$ will be:
$$H_2 = \begin{matrix} 8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{matrix}$$