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en.wikipedia.org/wiki/Logarithm_of_a_matrix#Existence mentions it always exists for all invertible complex matrices. But the real condition is more complicated. Notable counter example: -1 cannot be reached by any real .
The Lie algebra exponential covering problem can be seen as a generalized version of this problem, because
  • Lie algebra of is just the entire
  • we can immediately exclude non-invertible matrices from being the result of the exponential, because has inverse , so we already know that non-invertible matrices are not reachable

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