How Does a Convolution Can Be Expressed as a Matrix Multiplication (Matrix Form)?
For this operation to work, you need to imagine that your image is reshaped as a vector.
Then, this vector is multiplied on its left by the convolution matrix in order to obtain the blurred image.
Note that the result is also a vector the same size as the input, i.e., an image of the same size.
Each row of the convolution matrix corresponds to one pixel in the input image.
It contains the weight of the contributions of all the other pixels in the image to the blurred counterpart of the considered pixel.
Let's take an example: box blur of size $3 \times 3$ pixels on an image of size $6 \times 6$ pixels.
The reshaped image is a column of 36 elects, while the blur matrix has size $36 \times 36$.

Let's initialize this matrix to 0 everywhere.
Now, consider the pixel of coordinates $(i,j)$ in the input image (not on its border for simplicity). Its blurred counterpart is obtained by applying a weight of $1/9$ to itself and each of its neighbours at the positions $(i-1,j-1); (i-1,j), (i-1,j+1),\ldots,(i+1,j+1)$.
In the column vector, the pixel $(i,j)$ has the position $6*i + j$ (assuming lexicographic ordering). we report the weight $1/9$ in the $(6i+j)$-th line of the blur matrix.
Do the same with all other pixels.

A visual illustration of a closely related process (convolution + subtraction) can be found on my personal blog Computers Don't See (Yet) - Linearized model of LBPs.