Pfthb

A isthe kernel and B is an image. How do you find a convolution matrix out of this equation?

A(x,y) = B(x,y) + 4B(x+1,y-1) + 2B(x,y+1) + 5B(x-1,y)

And directions are as below

 (x-1,y-1) (x-1,y) (x-1,y+1)
 (x,y-1) (x,y) (x,y+1)
 (x+1,y-1) (x+1,y) (x+1,y+1)

is the matrix below?

 0 5 0
 0 1 2
 4 0 0

It depends on how you define your pixel coordinates. If the origin is at the right (!) bottom of the image, x runs from bottom to top and y from right to left, your matrix is correct. However this is quite an uncommon choice.

If your origin is at the bottom left, x runs from left to right and y runs from bottom to top, the matrix would be:

4 0 0
0 1 5
0 2 0

Note that the directions are inverted: For example, the matrix coefficient on the right of the center is applied to the picel on the left.

By the way, it is not correct that A is the kernel for arbitrary B. This is only the case for B[0,0] == 1 and B[x,y] == 0 for all other values of x and y.

Update:
So your x runs from top to bottom and your y from left to right. Then the convolution matrix is:

0 0 4
2 1 0
0 5 0