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
PhD student of Theoretical Particle Physics @UCIrvine l @NSF Fellow l Physics & Math Animations l Patreon: https://www.patreon.com/inertialobserver …
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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
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
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
Wait is L the derivative of R(θ) at θ=0 or is it -i times the derivative of R(θ) at θ=0? I think there’s a conflict between what you typed above and what’s in the picture...
It’s -i Times the derivative.. of you look closely since I raised e to +I they cancel out. You could argue that this is pointless, but the reason for that is so that L is hermitian as opposed to antihermitian.. this is a matter of convention
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