Matrix representation of loop transformations

To skew the inner loop by the outer loop by factor 1 we adjust the loop bounds, and replace I₁ by K₁, and I₂ by K₂-K₁. That is,

$$(K_1,K_2) = (I_1,I_2) \cdot {\bf U}$$

where U is a $2 \times 2$ matrix

$$\left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right]$$

That is,

$$(K_1,K_2) = (I_1,I_2) \cdot {\bf U} = (I_1,I_2+I_1)$$

The inverse gets us back again:

$$(I_1,I_2) = (K_1,K_2) \cdot {\bf U}^{-1} = (K_1,K_2-K_1)$$