〈 Berger | Dillon 〉

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