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Consider tranformation which turns one aligned rectangle to another:

enter image description here

This tranformation can be written in matrix form in the following way

enter image description here






i.e. transform matrix is diagonal.

My question is: what about arbitrary matrix? What 2 geometric figures defines it? Is this a pair of two arbitrary quadrangles?

If "yes", then what is the formula, relating angle coordinates to matrix coefficients?

How is it extrapolated to multidimensional?


I realized, that the number of equations in each transformation is $N$, the dimension of space, and the number of unknowns is $N^2+N=N(N+1)$ where $N$ is the number of translation column elements and $N^2$ is the number of linear matrix elements. Hence to make the system complete, we need $N(N+1)/N = (N+1)$ sample pairs of points.

So in the case of 2D we need to provide 2 arbitrary triangles to define transform.

Where to find a formula?

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