I think this part in article is starting point of the confusion :
See how the $I(x+u, y+v)$ changed into a totally different form
$I(x,y)+ uI_x + vI_y)$?
The author should have write a bit more detailed notation as:
$I(x,y)+ uI_x(x,y) + vI_y(x,y)$
i.e. pointing out that these are element-wise spatial derivatives (like Sobel) not matrix derivatives ( ...