Suppose I have two matrices A=[3 1;1 4] and B=[5 -2;-2 4]. Where A and B represent covariance ellipses in 2D. Now, if I want to combine it in 4D, that is, C=[A 0;0 B]. C is a four dimensional ellipsoid but the cross correlation between A and B is zero. I want to rotate A and B such that, they are highly correlated.
I have the following questions:
1) How can I rotate the A and B (either in 4D or 2D separately) such that they have maximum cross correlation with each others?
2) Is there any 4D rotation(either Euler or Quaternion) matrix in explicit form(I couldn't found any)?
Kindly guide me.
I'm sorry, but I can't really answer that one, but my instinct would be to first try a few empirical experiments. That is, to do a few calculations with differing starting values in order to see in which direction the answers move, and take it from there. That approach is often very productive in my experience.
Good luck!
Edward C. Kirby
The answer to 2) can be found in the english wikipedia
https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space#Group_structure_of_SO.284.29
see section "Relation to quaternions". Moreover, you can find some references there.
1) In my opinion this really depends on the properties of the two ellipses, and therefore, it will be hard to find a general approach.
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