Regression of a set of (2-element) vectors on another set of (2-element) vectors

Question

As a trivial example, here are two vectors:
f = {{1, 1}, {2, 2}, {3, 3}, {4, 4}}

s = {{-1, 2}, {-2, 4}, {-3, 6}, {-4, 8}}

It is instantly clear (apart from my sloppy formatting) that elements of s follow from elements of f according to s = M.f
where M = a matrix {{-1, 0}, {0, 2}}
I find the help details for the various forms of regression obscure, so my question is: more generally, how do I find the best fit matrix for such a (seemingly simple) correlation?

Daniel Huber · Answer

Note that your matrix M is not unique because your example data is not linear independent. Therefore, we may get a different M. Note further, due to the shape of your data you must write f.M not M.f.
To get a least square solution, we first define an symbolic matrix and define an error function: err that measures the differences between s and f.M. Then we minimize the error by adjusting the parameters of the symbolic matrix:
f = {{1, 1}, {2, 2}, {3, 3}, {4, 4}};
s = {{-1, 2}, {-2, 4}, {-3, 6}, {-4, 8}};
mat = Array[Subscript[m, #1, #2] &, {2, 2}];
err = (s - f.mat)^2 // Flatten // Total;
sol = mat /. FindMinimum[err, Flatten[mat]][[2]]
s == f.sol

(*{{-4.53077, 0.778831}, {3.53077, 1.22117}}*)
(*True*)

Regression of a set of (2-element) vectors on another set of (2-element) vectors

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