# Sobel and Prewitt operators decomposition

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?

The part: $$\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}$$ acts on vertical pixels $$\begin{bmatrix} x_{1,\cdot}\\ x_{2,\cdot}\\ x_{3,\cdot} \end{bmatrix}$$ as a weighted sum with equal weights, and if divided by $$3$$ like a 3-point average $$(x_{1,\cdot}+x_{2,\cdot}+x_{3,\cdot})/3$$.
Then, $$\begin{bmatrix} +1 & 0 & -1 \end{bmatrix}$$ acts on a row of horizontal pixels $$\begin{bmatrix} x_{\cdot,1}& x_{\cdot,2}& x_{\cdot,3} \end{bmatrix}$$ like the discrete 3-point derivative: $$(x_{\cdot,1}-x_{\cdot,3})/2$$ up to the $$2$$ factor. Here, up to a global $$6$$ factor, it acts like a smoother vertically combined with a gradient horizontally. The same reasoning works when you switch directions. This notion relates to the separability of a 2D mask into the product of two 1D operators.