I was reading in the wikipedia page of Sobel operator and Prewitt operator that is possible to decompose these two operators (quote form the Formulation paragraph):
"as the products of an averaging and a differentiation kernel, they compute the gradient with smoothing."
I know that this means that I can rewrite and simplify a Prewitt mask (for example) in this way:
\begin{align*} \begin{bmatrix} +1 & 0 & -1 \\ +1 & 0 & -1 \\ +1 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} \begin{bmatrix} +1 & 0 & -1 \end{bmatrix} \end{align*}
But I don't understand what are in these cases the averaging and the differentiation kernel? And, why it is written that they can compute the gradient with smoothing?