Computations in Grayscale morphology dilation
Structuring element is mostly supposed to be a binary array (though you can try 0 0 7 7 by yourself). H=[0 0 1 1]
u, v are the pixel coordinate at I. In your example, the only pixel value that will change is I(1,2) (the array label starts from 0 instead of 1).

I =  1     2     3
     3     7     2


Consider I(1,2), from the formula, I(1,2) I(1,1), and I(1,0) will be considered.

I(1,2) = max( I(1-0,2-0) + H(0,0), I(1-0,2-1) + H(0,1), ...
              I(1-0,2-2) + H(0,2), )
       = 7


That's the pixel value at (1,2) after dilation.

I will show you I(1,1):

I(1,1)=max(I(1-0,1-0)+H(0,0), I(1,0,1-1)+H(0,1))


That is still the original I(1,1).

EDIT

Mathematically, SE can has any values in its elements, but in Matlab, imdilate only accepts binary SE as a mask to determine the dilation region. You can still use [0 0 7 7] in your own calculation of dilation.

As your suggestion, I am showing you an example with SE of 3X3 size. I will calculate I(2,2).

I =  1     2     3 
     3     7     2
     3     7     2
H =  1     0     0
     0     1     1
     1     0     1

I(2,2) = max( I(2-0,2-0) + H(0,0), I(2-0,2-1) + H(0,1), I(2-0,2-2) + H(0,2),...
              I(2-1,2-0) + H(1,0), I(2-1,2-1) + H(1,1), I(2-1,2-2) + H(1,2),...
              I(2-2,2-0) + H(2,0), I(2-2,2-1) + H(2,1), I(2-2,2-2) + H(2,2))
       = 7