...

We can also represent loop interchange by a matrix transformation.

After transforming the skewed loop by matrix ${\bf V}=\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]$ (i.e. loop interchange) we get:

1.

Distance: (1,1), direction: (<,<)

2.

Distance: (1,0), direction: (<,.)

The transformed iteration space is the transpose of the skewed iteration space:

S00
S10	S11
S20	S21	S22
S30	S31	S32	S33
	S41	S42	S43
		S52	S53
			S63