Exponentially Better Rotations

Re:Exponentially Better Rotations by chombier on Tuesday April 19, 2022 @08:11AM (#31082056)

Thanks, I just realized I was also assuming exp to be surjective as well, I stand corrected.

The Wikipedia page[1] mentions the issue briefly but I was curious of the counter-examples. IIUC, matrices in SL(2) with trace < -2 have two distinct eigenvalues, one of which is negative[2], and such matrices cannot be reached by exponentiating elements of the Lie algebra sl(2) (traceless matrices).

[1] https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#S...

[2] https://en.wikipedia.org/wiki/SL2(R)#Classification_of_eleme...

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Re:Exponentially Better Rotations by chombier on Wednesday April 20, 2022 @04:19AM (#31094139)

I found this nice accessible proof for the interested: https://planetmath.org/slnrisconnected

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Re:Exponentially Better Rotations by andi999 on Thursday April 21, 2022 @11:49AM (#31111047)

Is this part of the proposition: x=exp X, and x having a double eigenvalue implying that X has a double eigenvalue somehow clear? How do you prove it?

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