Note that the last property 𝑅ⁿ(θ) = 𝑅(𝑛θ) is eerily similar to the "de Moivre property" .
This is indeed no accident
The underlying relationship is that the groups SO(2) and U(1) are 𝒊𝒔𝒐𝒎𝒐𝒓𝒑𝒉𝒊𝒄pic.twitter.com/KUFh4cRQN0
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Geometric intuition for what matrices do is your friend
Consider the 2D rotation matrix R(θ)
The Inverse of 𝑅(θ) is just "undoing" the rotation by θ
Rotating by θ 𝑛-times is the same thing as rotating once by 𝑛θpic.twitter.com/v0N9Af753F
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Here comes the real beauty
Define the matrix 𝐿 as the derivative of 𝑅(θ) about θ=0
𝑅(θ) can be written as a matrix exponential of 𝐿 (check it!)
We then obtain a beautiful matrix relation reminiscent of Euler's identity
This identity tells us that exp(𝑖𝐿θ) = 𝑅(θ)pic.twitter.com/zis6198nZq
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End of conversation
New conversation -
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This still works when you rotate around a random point instead origin
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