In Linear warping basics we saw how a basic warp/transformation can be represented with a matrix. To make more complex warpings (for example, to define a translation, rotation and scale as one warp) the individual matrices have to be multiplied through matrix multiplication. However matrix multiplication is not commutative, so the order of the set of matrices you use for the multiplication is going to be very important.

The first warping should be placed as the left-most matrix. The second warping to the right of that and so on. The second transformation is going to occur on the warped coordinates of the first. As an example for merging a few transforms into one matrix, the multiplication below represents the rotation of an image about a point \([\matrix{U&V}]\) anticlockwise from the horizontal axis by an angle of \(\theta\). To do this, first we take the origin to \([\matrix{U&V}]\) through translation. Then we rotate the image, then we translate it back to where it was initially. These three operations can be merged in one operation by calculating the matrix multiplication below:

$$\left[\matrix{1&0&U\cr0&1&V\cr{}0&0&1}\right] \left[\matrix{cos\theta&-sin\theta&0\cr sin\theta&cos\theta&0\cr 0&0&1}\right] \left[\matrix{1&0&-U\cr0&1&-V\cr{}0&0&1}\right]$$

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GNU Astronomy Utilities 0.22 manual, February 2024.